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Chapter 3: Problem 27

Use the graph of \(f\) to describe the transformation that yields the graph of \(g .\) $$f(x)=3^{x}, \quad g(x)=3^{x}+1$$

Short Answer

Expert verified
The transformation from the function \(f(x)=3^{x}\) to the function \(g(x)=3^{x}+1\) is a vertical shift of 1 unit upwards.

Step by step solution

01

Interpret the Given Functions

Two functions are given: \(f(x)=3^{x}\) and \(g(x)=3^{x}+1\). Here, \(f(x)\) is the original function and \(g(x)\) is the transformed function.
02

Identify the Transformation

We can see that \(g(x)\) is equal to \(f(x)\) with a '+1' added. This '+1' is not inside the term with \(x\) or multiplied with \(x\), but rather it's added separately. Therefore this suggests a vertical shift.
03

Determine the Direction and Magnitude of the Shift

Since the '+1' is added to the original function, the graph is shifted upwards. The magnitude of the shift is 1, which is the same as the number added to the original function.

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Most popular questions from this chapter

True or False? In Exercises 83 and \(84,\) determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{6} x\) is a reflection of the graph of \(g(x)=6^{x}\) in the \(x\) -axis.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Write the logarithmic equation in exponential form. $$\ln 7=1.945 \ldots$$

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph.\( \)y=\log (-x)$$

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