Questions: For f(x)=4 x+1 and g(x)=x^2-5, find (g/f)(x). A. (x^2-5)/(4 x+1), x ≠ -1/4 B. (x^2-5)/(4 x+1) C. (4 x+1)/(x^2-5) D. (4 x+1)/(x^2-5), x ≠ ± sqrt(5)

Transcript text: For $f(x)=4 x+1$ and $g(x)=x^{2}-5$, find $\left(\frac{g}{f}\right)(x)$. A. $\frac{x^{2}-5}{4 x+1}, x \neq-\frac{1}{4}$ B. $\frac{x^{2}-5}{4 x+1}$ C. $\frac{4 x+1}{x^{2}-5}$ D. $\frac{4 x+1}{x^{2}-5}, x \neq \pm \sqrt{5}$

Solution

Solution Steps

Step 1: Define the Functions

We are given two functions: \[ f(x) = 4x + 1 \] \[ g(x) = x^2 - 5 \]

Step 2: Calculate $\left(\frac{g}{f}\right)(x)$

To find $\left(\frac{g}{f}\right)(x)$, we compute: \[ \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{x^2 - 5}{4x + 1} \]

Step 3: Simplify the Expression

The expression $\left(\frac{g}{f}\right)(x)$ is already in its simplest form: \[ \left(\frac{g}{f}\right)(x) = \frac{x^2 - 5}{4x + 1} \]

Step 4: Determine Domain Restrictions

The expression is undefined when the denominator is zero. We set the denominator equal to zero: \[ 4x + 1 = 0 \implies x = -\frac{1}{4} = -0.25 \] Thus, the expression is undefined for $x = -0.25$.