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Chapter 1: Problem 54

Write an equation for the function described by the given characteristics. The shape of \(f(x)=\sqrt{x},\) but shifted nine units down and then reflected in both the \(x\) -axis and the \(y\) -axis

Short Answer

Expert verified
The function is \(h(x) = 9 - \sqrt{-x}\) for \(x \le 0\).

Step by step solution

01

Shift Nine Units Down

Shifting a function nine units down effectively translates to subtracting 9 from each y-coordinate of the function. For the function \(f(x)=\sqrt{x}\), let's denote \(g(x)\) as \(g(x) = \sqrt{x} - 9\).
02

Reflect Along the X-Axis

Reflecting a function along the x-axis means changing the sign of the y-coordinate. Therefore, after the reflection, say \(\varphi(x)\), we get \(\varphi(x) = -g(x) = -(\sqrt{x} - 9)\). Simplifying this gives us \(\varphi(x) = 9 - \sqrt{x}\).
03

Reflect Along the Y-Axis

Reflecting a function along the y-axis means changing the sign of the x-coordinate. Therefore, after the reflection, let's denote the final function as \(h(x)\), we get \(h(x) = \varphi(-x) = 9 - \sqrt{-x}\). We should swiftly note that the square root of a negative number is not a real number. Therefore, this function is not continuous and it has its domain limited to \(x \le 0\).

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Most popular questions from this chapter

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$h(x)=\sqrt{x+2}+3$$

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