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Chapter 1: Problem 6

Assume \(f(x)=\frac{x+2}{x^{2}+1}\) for every real number \(x .\) Evaluate and simplify each of the following expressions. \(f\left(\frac{b}{3}\right)\)

Short Answer

Expert verified
\(f\left(\frac{b}{3}\right) = \frac{3(b+6)}{b^2 + 9}\)

Step by step solution

01

Substitute the input into the function

Replace \(x\) with \(\frac{b}{3}\) in the function f(x). So, we'll have \(f\left(\frac{b}{3}\right) = \frac{\frac{b}{3} + 2}{(\frac{b}{3})^2 + 1}\)
02

Simplify the numerator

To simplify the numerator, we'll combine the terms: \[\frac{b}{3} + 2 = \frac{b+6}{3}\] So, the expression becomes: \(f\left(\frac{b}{3}\right) = \frac{\frac{b+6}{3}}{(\frac{b}{3})^2 + 1}\)
03

Simplify the denominator

To simplify the denominator, we'll square the term \(\frac{b}{3}\) and add 1: \[(\frac{b}{3})^2 + 1 = \frac{b^2}{9} + 1\] To combine the terms, we'll need to find a common denominator, which is 9: \[\frac{b^2}{9} + 1 = \frac{b^2 + 9}{9}\] So, the expression becomes: \(f\left(\frac{b}{3}\right) = \frac{\frac{b+6}{3}}{\frac{b^2 + 9}{9}}\)
04

Divide the fractions

To divide the fractions, we'll multiply the first fraction by the reciprocal of the second fraction: \[\frac{\frac{b+6}{3}}{\frac{b^2 + 9}{9}} = \frac{b+6}{3} \times \frac{9}{b^2 + 9}\]
05

Simplify the resulting expression

Now, multiply the numerators and denominators: \[\frac{b+6}{3} \times \frac{9}{b^2 + 9} = \frac{(b+6)(9)}{3(b^2 + 9)}\] And then we can simplify the expression by dividing both the numerator and the denominator by 3: \[\frac{(b+6)(9)}{3(b^2 + 9)} = \frac{(b+6)(3)}{(b^2 + 9)}\] Finally, we get: \(f\left(\frac{b}{3}\right) = \frac{3(b+6)}{b^2 + 9}\)

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Most popular questions from this chapter

Suppose \(g\) is the function whose domain is the interval [-2,2] , with \(g\) defined on this domain by the formula $$g(x)=\left(5 x^{2}+3\right)^{7777}$$ Explain why \(g\) is not a one-to-one function.

Check your answer by evaluating the appropriate function at your answer. Suppose \(f(x)=3 x+2\). Find a formula for \(f^{-1}\).

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=-5 f\left(-\frac{4}{3} x\right)-8\)

Suppose \(f\) is a function whose domain equals \\{2,4,7,8,9\\} and whose range equals \(\\{-3,0,2,6\\} .\) Explain why \(f\) is not a one-to-one function.

For each of the functions \(f\) given. (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part (c) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I\) (recall that \(I\) is the function defined by \(I(x)=x\) ). \(f(x)=3 x+5\)
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