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Chapter 1: Problem 8

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

Short Answer

Expert verified
The simplified expression for \(f(3a - 1)\) is: \[f(3a - 1) = \frac{3a + 1}{9a^2 - 6a + 2}\]

Step by step solution

01

Substitute the input into the function

We are asked to find \(f(3a - 1)\). So we will substitute (3a - 1) for x in the function. \[f(3a - 1) = \frac{3a - 1 + 2}{(3a - 1)^2 + 1}\]
02

Simplify the numerator

Now, we will simplify the numerator by combining the constants (-1 and 2). \[f(3a - 1) = \frac{3a + 1}{(3a - 1)^2 + 1}\]
03

Simplify the denominator

In the denominator, we must expand the square and add 1 to the result. \[(3a - 1)^2 + 1 = (3a - 1)(3a - 1) + 1 = 9a^2 - 6a + 1 + 1 = 9a^2 - 6a + 2\] Now we have to substitute the simplified denominator back into the expression, \[f(3a - 1) = \frac{3a + 1}{9a^2 - 6a + 2}\]
04

Present the final simplified expression

The simplified expression for \(f(3a - 1)\) is: \[f(3a - 1) = \frac{3a + 1}{9a^2 - 6a + 2}\]

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