WEBVTT
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In this video, we will learn how to calculate the value or output of a function using its equation or graph.
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We will begin by recalling what we mean by a function.
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And then we will explain how we can evaluate them.
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The most popular function notation is π of π₯.
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The π of π₯ notation is another way of representing the π¦-value in a function such that π¦ is equal to π of π₯.
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Instead of writing π¦ equals three π₯ plus seven, we will write π of π₯ is equal to three π₯ plus seven.
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The π¦-axis may be labeled as the π-of-π₯ axis when graphing the function.
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Labeling a function in this way avoids confusion when weβre dealing with multiple functions.
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We will now look at how we can evaluate a function.
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To evaluate a function, we substitute the input or given number for the functionβs variable, usually π₯.
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For example, letβs consider the function π of π₯ is equal to three π₯ plus seven.
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If we were asked to evaluate π of four, then four is the input.
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We substitute this into the expression.
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In this case, we have three multiplied by four plus seven.
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This is equal to 19, which is known as the output.
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When the input to the function is four, the output is 19.
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This can be written as an ordered pair or coordinate four, 19, where four is the π₯-value and 19 is the π-of-π₯ or π¦-value.
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We will now look at some questions that involve evaluating functions.
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Using the function π¦ equals π₯ squared plus three, calculate the corresponding output for an input of two.
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The output of any function in this format is the π¦-value.
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The input is the π₯-value.
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We need to substitute π₯ is equal to two into the function.
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As π¦ is equal to π₯ squared plus three, when π₯ is two, π¦ is equal to two squared plus three.
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Two squared is equal to four.
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So π¦ is equal to four plus three.
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As this is equal to seven, the output that corresponds to an input of two is seven.
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We could write this as an ordered pair with coordinates two, seven.
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The point with coordinates two, seven lies on the quadratic function π¦ equals π₯ squared plus three.
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When dealing with functions, the π¦ is often replaced with π of π₯.
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These are interchangeable, and both correspond to the output.
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Our next question involves completing a table of values for a function.
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Complete the table of values for the function π¦ equals three π₯ squared minus two π₯.
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In this question, weβre given a function π¦ equals three π₯ squared minus two π₯ and five integer values of π₯ from negative two to two.
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In order to complete the table, we need to substitute each of these values in turn into the function.
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Letβs begin with positive two.
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When our π₯-value or input is equal to two, then our π¦-value or output will be equal to three multiplied by two squared minus two multiplied by two.
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Using our order of operations, three multiplied by two squared is equal to 12.
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We need to square the two and then multiply by three.
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Two multiplied by two is four.
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So we have 12 minus four.
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This is equal to eight.
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When π₯ is equal to two, π¦ is equal to eight.
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We can repeat this process when π₯ is equal to one.
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Three multiplied by one squared is three, and two multiplied by one is two.
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As three minus two is equal to one, when π₯ is equal to one, π¦ is equal to one.
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When π₯ is equal to zero, π¦ is equal to three multiplied by zero squared minus two multiplied by zero.
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Both parts of this calculation are equal to zero.
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And zero minus zero is zero.
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We now need to consider when π₯ is negative, which is slightly more complicated.
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Squaring a negative number gives a positive answer, as multiplying a negative by a negative is a positive.
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This means that three multiplied by negative one squared is equal to three.
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Two multiplied by negative one is negative two.
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But as weβre subtracting this, weβre left with positive two.
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Three plus two is equal to five.
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So when π₯ is equal to negative one, π¦ is equal to five.
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Three multiplied by negative two squared is 12, as negative two squared is four.
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As two multiplied by negative two is negative four, we need to add four to 12.
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Once again, weβre subtracting a negative number.
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This gives us an output or π¦-value of 16 when π₯ is negative two.
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The five missing values in the table are 16, five, zero, one, and eight.
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We could use these coordinate pairs negative two, 16; negative one, five; and so on to graph the function π¦ equals three π₯ squared minus two π₯.
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As our function is quadratic and the coefficient of π₯ squared is positive, we will have a U-shaped parabola.
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In our next question, we will need to identify which point satisfies a given function.
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Which of the following set of coordinates lies on π of π₯ is equal to negative 19π₯ minus 16?
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Is it (A) 10, negative 16; (B) 10, negative 206; (C) negative 206, 10; or (D) negative 206, negative 16?
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We are used to any pair of coordinates being written in the form π₯, π¦.
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The first coordinate is the π₯-value, and the second is the π¦-value.
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The function π of π₯ is equal to negative 19π₯ minus 16 is the same as π¦ equals negative 19π₯ minus 16.
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As π¦ and π of π₯ are interchangeable, we can write the coordinate pair as π₯, π of π₯.
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The first value is known as the input and the second value the output.
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We need to work out which of the four options satisfies the function.
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We notice that both option (A) and option (B) have an input or π₯-value of 10.
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This means that in order to calculate the output, we need to work out π of 10.
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Substituting in π₯ equals 10 gives us negative 19 multiplied by 10 minus 16.
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Multiplying a negative number by a positive gives a negative answer.
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So negative 19 multiplied by 10 is negative 190.
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Subtracting 16 from this gives us negative 206.
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This means that the coordinate pair 10, negative 206 lies on the function.
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As option (A) was 10, negative 16, this is incorrect.
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Option (B), on the other hand, was 10, negative 206.
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So this is the correct answer.
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We could check to see if options (C) and (D) are correct by substituting negative 206 in for π₯.
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π of negative 206 is equal to negative 19 multiplied by negative 206 minus 16.
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We can clearly see that this will not satisfy either of our options, as our value is too large.
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An input of negative 206 gives an output of 3898.
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The coordinate pair negative 206, 3898 lies on the function.
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This does not correspond to option (C) or (D).
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So these are both incorrect.
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The correct answer is option (B).
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Our next example is a multipart question involving the meaning of a function.
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Given the function π, the meaning of π of π minus one is the output when the input is one less than π.
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Interpret the following.
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π of π plus three, π of π minus three, π of three minus π₯, π of π minus π of π, π of three π‘, and π of π to the power π.
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Before starting this question, it is worth recalling what we mean by a function π.
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If we have any function π of π₯, then π₯ is the input and π of π₯ is the output.
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A number inside the bracket affects the input, whereas a number outside of the bracket affects the output.
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This can be seen from the example, as π minus one means one less than π.
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Our first function, π of π plus three, is very similar to the example.
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Instead of subtracting one from π, weβre adding three to π.
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This means that π of π plus three calculates the output when the input is three more than π.
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Our second function, π of π minus three, is slightly different.
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This time, the three that is being subtracted is outside of the bracket.
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π of π will be the output when the input is π .
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Therefore, π of π minus three is three less than the output when the input is π .
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Our third function, π of three minus π₯, is very similar to the first one and also the example.
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This time, our function gives us the output when the input is π₯ less than three.
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We are subtracting π₯ from three and then working out the output.
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Our fourth function has two variables, π and π.
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We have π of π minus π of π.
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This means that we are subtracting the value of π of π from the value of π of π.
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This is the difference between them.
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Therefore, the answer corresponds to the change in output when the input changes from π to π.
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The penultimate function is π of three π‘.
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We are multiplying our value of π‘ by three.
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This is a similar idea once again to our first function, π of π plus three, and also our third function, π of three minus π₯.
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This time, it corresponds to the output when the input is three times π‘.
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Our final function involves exponents or powers.
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We have π of π to the power of π.
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As the π is outside of the bracket or parentheses, this is the result of raising the output at input π to the πth power.
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If the power π was inside the bracket, we would be raising the input to the πth power.
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Interpreting and often drawing functions of this type is an important part of the topic.
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Our final question involves evaluating a function from a graph.
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Determine π of one.
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We can see from the graph that our axes are labeled π₯ and π of π₯.
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Any function written in the form π of π₯ has input π₯ and output π of π₯.
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In this question, our value of π₯ or our input is one.
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We need to find the value of π of π₯ from the graph when π₯ is equal to one.
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We do this by drawing a vertical line from one on the π₯-axis.
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Once we reach our graph, we draw a horizontal line across to the π¦- or π-of-π₯ axis.
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This is equal to eight.
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Therefore, the value of π of one is eight.
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We could work out the value of π of negative two up to π of eight using this graph.
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We will now finish this video by summarizing the key points.
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As mentioned at the start of the video, the π of π₯ notation is another way of representing the π¦-value of a function such that π¦ is equal to π of π₯.
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If we consider the function π of π₯ equals five π₯ minus two, then our π₯-value is the input.
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The value of five π₯ minus two is the output value.
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To evaluate a function, we substitute the input for the functionβs variable.
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For example, to calculate π of three, we substitute three for π₯.
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Five multiplied by three minus two is 13.
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Therefore, the input of three gives an output of 13.
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We have seen from this video that we can calculate an output using equations, sometimes in the form of a table, or alternatively from graphs.