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Chapter 3: Problem 28

Use the graph of \(f\) to describe the transformation that yields the graph of \(g .\) $$f(x)=10^{x}, \quad g(x)=10^{-x+3}$$

Short Answer

Expert verified
The graph of \(g(x) = 10^{-x+3}\) is obtained by reflecting the graph of the basic function \(f(x) = 10^x\) about the y-axis and then shifting it 3 units to the right.

Step by step solution

01

Understand the basic function

The given function \(f(x) = 10^x\) is a basic exponential function, where the base is 10. The graph of \(f\) is a smooth curve that passes through the point (0, 1), increases for \(x > 0\) and tends towards 0 for \(x < 0\). It is unbounded above and bounded below by the x-axis.
02

Understand the transformation

The function \(g(x) = 10^{-x+3}\) can be rewritten as \(g(x) = 10^{-(x-3)}\). Here, compared to the basic function \(f\), the function \(g\) is the result of replacing \(x\) with \(-(x-3)\) in \(f\). This transformation can be broken down into two parts: a reflection about the y-axis because of the negative sign, and a horizontal shift to the right by 3 units due to the '-3' inside the brackets. Essentially, every point \((x, y)\) on the graph of \(f\) is transformed to \((-x + 3, y)\) on the graph of \(g\).
03

Piece together the transformation

Putting together the reflections and translations, the graph of \(g\) is obtained from the graph of \(f\) by reflecting the graph of \(f\) about the y-axis and then shifting it 3 units to the right.

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Most popular questions from this chapter

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Use a graphing utility to graph the functions \(y_{1}=\ln x-\ln (x-3)\) and \(y_{2}=\ln \frac{x}{x-3}\) in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.

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