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Chapter 1: Problem 18

Evaluate the indicated function for \(f(x)=x^{2}+1\) and \(g(x)=x-4\). $$ (f-g)(-1) $$

Short Answer

Expert verified
The result of (f - g)(-1) is 7.

Step by step solution

01

Defining the Functions

Two functions are given here. The first function is \(f(x) = x^{2} + 1\) and the second function is \(g(x) = x - 4\). They are to be used in the operation (f - g)(-1).
02

Performing the Subtraction

First, perform the operation f - g. Doing this yields a new function. This is done by subtracting 'x - 4' from 'x^2 + 1'. It looks like this: \(h(x) = f(x) - g(x) = (x^2 + 1) - (x - 4) = x^2 + 1 - x + 4\). One will then be able to tidy the equation to result in \(h(x) = x^2 - x + 5\). This is our function (f - g).
03

Evaluating the Function

Now, for the final step, one must evaluate the new function (f - g) at the indicated value of x, which is -1 in this case. So, substitute -1 for x: \(h(-1) = (-1)^2 - (-1) + 5 = 1 + 1 + 5 = 7\).

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Most popular questions from this chapter

The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) $$\begin{array}{llllll} 1920 & 146.6 & 1956 & 184.9 & 1984 & 218.5 \\ 1924 & 151.3 & 1960 & 194.2 & 1988 & 225.8 \\ 1928 & 155.3 & 1964 & 200.1 & 1992 & 213.7 \\ 1932 & 162.3 & 1968 & 212.5 & 1996 & 227.7 \\ 1936 & 165.6 & 1972 & 211.3 & 2000 & 227.3 \\ 1948 & 173.2 & 1976 & 221.5 & 2004 & 229.3 \\ 1952 & 180.5 & 1980 & 218.7 & 2008 & 225.8 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012 .

Determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$

Use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ g^{-1} \circ f^{-1} $$

The function given by \(y=0.03 x^{2}+245.50, \quad 0

Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x) .\)
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