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Chapter 1: Problem 53

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

Short Answer

Expert verified
The equation of the function described by the given characteristics is \(f(x) = -\sqrt{-x+6}\).

Step by step solution

01

Understand the Shift Transformation

The description mentions a shift of six units to the left. In function terms, a horizontal shift of \(h\) units to the left is represented by replacing every \(x\) in the function with \(x+h\). So, the transformed function becomes \(f(x)=\sqrt{x+6}\).
02

Understand the Reflection Transformation

The description mentions a reflection in both the x- and y-axes. A reflection around the x-axis is achieved by replacing \(f(x)\) with \(-f(x)\), and a reflection around the y-axis is achieved by replacing each \(x\) in the function with \(-x\). So, apply these transformations one by one to the current function. Reflecting around the x-axis gives \(-f(x)=-\sqrt{x+6}\), and then reflecting around the y-axis gives \(-f(-x)=-\sqrt{-x+6}\).
03

Finalize the Equation

Combining all transformations, the final equation for the transformed function is \(f(x)=-\sqrt{-x+6}\). This function has the same shape as \(f(x)=\sqrt{x}\), but it's shifted six units to the left and reflected in both the x- and y-axes.

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