Question
\( f(x)=3 x-1 \) \( g(x)=x^{2}-x \) Find \( \left(\frac{f}{g}\right)(x) \)
Ask by Perkins Montgomery. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\left(\frac{f}{g}\right)(x) = \frac{3x - 1}{x(x - 1)}
\]
Solution
To find \(\left(\frac{f}{g}\right)(x)\), you simply divide \(f(x)\) by \(g(x)\).
Given:
\[
f(x) = 3x - 1
\]
\[
g(x) = x^2 - x
\]
So,
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 1}{x^2 - x}
\]
This can be further factored in the denominator:
\[
x^2 - x = x(x - 1)
\]
Thus, the simplified form is:
\[
\left(\frac{f}{g}\right)(x) = \frac{3x - 1}{x(x - 1)}
\]
**Final Answer:**
\[
\left(\frac{f}{g}\right)(x) = \frac{3x - 1}{\,x(x - 1)}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find \(\left(\frac{f}{g}\right)(x)\), we need to compute the ratio of the two functions \(f(x)\) and \(g(x)\). 1. We start by substituting the functions into the ratio: \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 1}{x^2 - x} \] 2. The next step is to ensure the denominator is not zero. The function \(g(x) = x^2 - x\) can be factored: \[ g(x) = x(x - 1) \] So, \(g(x) = 0\) when \(x = 0\) or \(x = 1\). Thus, the final result is: \[ \left(\frac{f}{g}\right)(x) = \frac{3x - 1}{x(x - 1)}, \quad \text{for } x \neq 0 \text{ and } x \neq 1. \]
