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Chapter 1: Problem 63

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=(x+3)^{2}, \quad x \geq-3$$

Short Answer

Expert verified
The function \( f(x) = (x+3)^{2} \), with \( x \geq -3 \), does have an inverse. Its inverse is \( f^{-1}(x) = \sqrt{x} - 3 \).

Step by step solution

01

Check if the function has an inverse

For a function to have an inverse, it needs to be one-to-one, meaning that each input x corresponds to a unique output y. This function is one-to-one within its restricted domain of \( x \geq -3 \). Therefore, it does have an inverse.
02

Find the inverse function

To find the inverse function, replace \( f(x) \) with \( y \), and then swap \( y \) and \( x \). Solve for \( y \). This gives the equation \( x = (y + 3)^{2} \). Find the square root of both sides to get \( y = \sqrt{x} - 3 \). Though, usually when we square root we consider both positive and negative values, here we consider only positive since the domain of x was \( \geq -3 \). Inverse of \( y \) is denoted by \( f^{-1}(x) \). So, \( f^{-1}(x) = \sqrt{x} - 3 \).

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