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Consider the function π of π₯ equals the cubed root of 125 minus the square root of two π₯ plus three.
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Part (a): Find the domain of π of π₯.
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Part (b): Find the range of π of π₯.
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We have here a cube root function and then a square root function in the expression within the cube root.
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We recall first that the domain of a cube root function is the set of all real numbers.
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So as there is no restriction on the domain of the cube root function, we only need to consider the restriction for the square root function.
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The domain of a square root function must be nonnegative.
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So we have the inequality two π₯ plus three is greater than or equal to zero.
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Solving this inequality leads to π₯ is greater than or equal to negative three over two.
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So we find that the domain of the function π of π₯ is the left-closed, right-open interval from negative 1.5 to β, because we can only evaluate the entire expression under the cube root when π₯ is in this interval.
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Letβs now consider the range of π of π₯.
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We can write our function as π of π₯ equals the cube root of 125 minus π, where π is equal to the square root of two π₯ plus three.
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We know that the range of a square root function is the set of all nonnegative values.
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And so π is greater than or equal to zero.
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The largest value of π of π₯ will correspond to the smallest value of π, which is zero.
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So the largest value of π of π₯ is the cube root of 125 minus zero, which is the cube root of 125, which is five.
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The smallest value of π of π₯ will correspond to the largest value of π.
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So as π tends to β, π of π₯ will tend to the cube root of negative β, which itself tends to negative β.
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As π of π₯ is a continuous function, its range will therefore include all the values between its smallest and largest value, which is the left-open, right-closed interval from negative β to five.
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So weβve completed the problem and found the domain and range of this fairly complicated radical function, which involves both square and cube roots.
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The domain is the left-closed, right-open interval from negative 1.5 to β.
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And the range is the right-open, left-closed interval from negative β to five.