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Chapter 3: Problem 18

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=\left(\frac{1}{2}\right)^{-x}$$

Short Answer

Expert verified
The graph of the function \(f(x)=(\frac{1}{2})^{-x}\) shows a typical exponential decay with y-values quickly increasing as x-values get more negative and quickly decreasing towards 0 as x-values get more positive.

Step by step solution

01

Understand the Function

The function provided is \(f(x)=(\frac{1}{2})^{-x}\). It is an exponential function where the base is \(\frac{1}{2}\) and the exponent is \( -x \). By nature, exponential functions occur when the variable is in the exponent and a constant is the base. Here, the negative exponent implies the reciprocal of the base.
02

Generate the Table of Values

With the help of a graphing utility, input the function and generate a table of values. For instance: when \(x=-2\), \(f(x)=4\); when \(x=-1\), \(f(x)=2\); when \(x=0\), \(f(x)=1\); when \(x=1\), \(f(x)=0.5\); when \(x=2\), \(f(x)=0.25\) etc. Remember, in an exponential function, as x increases, the \(f(x)\) values tend to diminish and vice versa.
03

Sketch the Graph

Use the generated table of values to sketch the graph of the function. The graph should pass through the points corresponding to the x and \(f(x)\) values in the table. For an exponential function as this, the graph should show a rapid increase in y-values for negative x-values and a rapid decrease towards 0 without touching the x-axis for positive x-values.

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Most popular questions from this chapter

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). .\(r\)\ln \left(x^{2}-2\right)=\ln 23$

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). $$\ln (x-7)=\ln 7$$

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-4}$$

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{-0.9}=0.406 \ldots$
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