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Chapter 1: Problem 64

Determine whether the function has an inverse function. If it does, then find the inverse function. $$q(x)=(x-5)^{2}$$

Short Answer

Expert verified
The function q(x) does not have an inverse over the entire real line. However, if its domain is restricted to x >= 5, then the inverse exists and it is q^-1(y) = sqrt(y) + 5.

Step by step solution

01

Demonstrate that q(x) is one-to-one on a restricted domain

Let's show that q(x) is one-to-one on the domain of x>=5. With x1, x2 in [5,鈭�), if q(x1) = q(x2), then (x1-5)^2 = (x2-5)^2. The square root function is defined for non-negative values, so we can take the square root of both sides to get |x1-5| = |x2-5|. For x1, x2 >=5, |x-5| = x-5, thus x1 = x2. So q(x) is one-to-one on x>=5.
02

Demonstrate that q(x) is onto on the range

Since q(x) = (x-5)^2, its range (values of y) spans all non-negative real numbers [0, 鈭�), so q(x) is onto.
03

Determine the inverse function q^-1(y)

Once a bijective function q(x) is defined on a particular domain, we can find its inverse q^-1(y) by switching the roles of y and x, yielding y = (x-5)^2, and then solving for x to get x = y^(1/2) + 5. Therefore, q^-1(y) = sqrt(y) + 5.

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