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Chapter 1: Problem 11

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

Short Answer

Expert verified
The short answer to the given expression is \(f\left(\frac{a}{b} - 1\right) = \frac{\frac{a}{b} + 1}{\frac{a^2}{b^2} - \frac{2a}{b} + 2}\).

Step by step solution

01

Substitute the Given Expression into the Function

Make the substitution \(x = \frac{a}{b} - 1\) into the given function \(f(x) = \frac{x+2}{x^2+1}\): \(f\left(\frac{a}{b} - 1\right) = \frac{\left(\frac{a}{b} - 1\right) + 2}{\left(\frac{a}{b} - 1\right)^2 + 1}\)
02

Simplify the Expression in the Numerator

Next, simplify the numerator of the function: \(=\frac{\frac{a}{b} + 1}{\left(\frac{a}{b} - 1\right)^2 + 1}\)
03

Simplify the Expression Inside the Parenthesis in the Denominator

Simplify the expression inside the parenthesis in the denominator: \(\left(\frac{a}{b} - 1\right)^2 = \frac{a^2}{b^2} - \frac{2a}{b} + 1\)
04

Simplify the Denominator With the Squared Expression

Substitute the simplified squared expression back into the denominator and add 1: \(\frac{a^2}{b^2} - \frac{2a}{b} + 1 + 1 = \frac{a^2}{b^2} - \frac{2a}{b} + 2\)
05

Combine Numerator and Denominator

Put the simplified numerator and denominator together to get the final simplified expression: \(f\left(\frac{a}{b} - 1\right) = \frac{\frac{a}{b} + 1}{\frac{a^2}{b^2} - \frac{2a}{b} + 2}\)

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Most popular questions from this chapter

Suppose \(f\) and \(g\) are functions. Show that the composition \(f \circ g\) has the same domain as \(g\) if and only if the range of \(g\) is contained in the domain of \(f\).

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

Use the U. S. 2011 federal income tax function for a single person as defined in Example 2 of Section 1.1. What is the taxable income of a single person who paid $$\$ 20,000$$ in federal taxes for \(2011 ?\)

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ What is the domain of \(g^{-1} ?\)

Show that the sum of two increasing functions is increasing.
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