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Consider the functions π of π₯ equals π₯ minus two and π of π₯ equals five minus π₯ squared.
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Part a) Show that π of π of π₯ equals one plus four π₯ minus π₯ squared.
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π of π of π₯ is a composite function.
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It means that we take an input value π₯, apply the function π first, and then apply π to the result.
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In this case though, weβre looking to find a general algebraic expression for the composite function π of π of π₯.
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We start with an π₯-value.
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The function π of π₯ is π₯ minus two, meaning we subtract two from our π₯-value.
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This then becomes the input for the second function, π.
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The function π of π₯ is five minus π₯ squared.
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So we take our new input of π₯ minus two and replace π₯ with it, giving five minus π₯ minus two all squared.
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We need to simplify this algebraic expression.
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So first, weβll expand the brackets.
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π₯ minus two all squared means π₯ minus two multiplied by π₯ minus two.
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You can use the FOIL method to help with this expansion if you wish.
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π₯ multiplied by π₯ gives π₯ squared.
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π₯ multiplied by negative two gives negative two π₯.
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Negative two multiplied by π₯ gives another lot of negative two π₯.
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And negative two multiplied by negative two gives positive four.
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We can then simplify this expansion by grouping like terms.
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In the centre, we have negative two π₯ minus two π₯, which is equal to negative four π₯.
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So we can replace π₯ minus two all squared with our expanded version of π₯ squared minus four π₯ plus four.
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Now we need to expand this larger bracket which has a negative sign in front of it.
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This means that weβre multiplying the whole bracket by negative one, which just has the effect of changing all of the signs.
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So the five stays the same.
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We then have negative π₯ squared plus four π₯ minus four.
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Weβre nearly there with the simplification.
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But we just have one final step, which is to group the like terms.
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In this case, the only like terms are the five and negative four, which are both constant terms.
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They have no π₯s or π₯ squareds.
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Five minus four is one.
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And weβll just swap the order of the other two terms around to be consistent with the way π of π of π₯ is written in the question.
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Weβve shown then that π of π of π₯ is equal to one plus four π₯ minus π₯ squared as required.
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Part b) of the question says, βSolve π of π of π₯ equals π of π of π₯.β
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Here the two functions have been composed in different orders.
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π of π of π₯, remember, means we apply π first and then apply π, whereas π of π of π₯ means we apply π first and then apply π.
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Applying functions in different orders does not give the same result in general.
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We have an algebraic expression for π of π of π₯ from part a).
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Itβs one plus four π₯ minus π₯ squared.
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But we donβt have one for π of π of π₯.
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So our first step is going to be to find this composite function.
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π of π₯ is the function five minus π₯ squared, and then weβre applying π to this.
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π of π₯ is the function π₯ minus two, meaning we take our input value and subtract two.
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So if our input value is five minus π₯ squared, then we now have five minus π₯ squared minus two.
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This simplifies to three minus π₯ squared.
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So this is our algebraic expression for π of π of π₯.
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Next, weβll take our two algebraic expressions for π of π of π₯ and π of π of π₯ and substitute them into the two sides of the equation.
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π of π of π₯ is one plus four π₯ minus π₯ squared, and π of π of π₯ is three minus π₯ squared.
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So we have the equation one plus four π₯ minus π₯ squared equals three minus π₯ squared.
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Notice that both sides of this equation have a term of negative π₯ squared.
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So these will cancel each other out.
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You could also see this by adding π₯ squared to each side of the equation.
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Once these terms have been cancelled out, weβre left with the linear equation one plus four π₯ equals three, which we need to solve for π₯.
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To do so, we first subtract one from each side, giving four π₯ equals two, and then divide both sides of the equation by four, giving π₯ equals two over four.
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But of course, the fraction two over four should be simplified to one over two or one-half by dividing both the numerator and denominator by two.
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So weβve solved the equation π of π of π₯ equals π of π of π₯ and found that π₯ is equal to a half.
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You could of course substitute this value of a half back into the two composite functions and confirm that they do indeed give the same answer.