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Chapter 1: Problem 21

Evaluate (if possible) the function at each specified value of the independent variable and simplify. \(f(x)=2 x-3\) (a) \(f(1)\) (b) \(f(-3)\) (c) \(f(x-1)\)

Short Answer

Expert verified
After evaluating, the results obtained are: \(f(1)=-1\), \(f(-3)=-9\) and \(f(x-1)=2x-5\)

Step by step solution

01

Evaluate \(f(1)\)

Substitute `1` into the function in place of `x` to obtain \(f(1)=2*1-3\), which simplifies to \(f(1)=-1\).
02

Evaluate \(f(-3)\)

Substitute `-3` into the function in place of `x` to obtain \(f(-3)=2*-3-3\). This simplifies to \(f(-3)=-9\).
03

Evaluate \(f(x-1)\)

Substitute `x-1` into the function in place of `x` to obtain \(f(x-1)=2*(x-1)-3\). This simplifies to \(f(x-1)=2x-5\) after applying the distributive property of multiplication over subtraction and combining similar terms.

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Most popular questions from this chapter

(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f(-5)=-1, \quad f(5)=-1$$

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=x^{3}-1$$

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=|x-1|$$
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