Questions: Let g(x) = (x-1)/2 and f(x) = 4x-1 then find g(f(x)) A) 2x-1 B) 2x C) 16x^2+1 D) (x-1)/4 E) None of these

Transcript text: Let $g(x)=\frac{x-1}{2}$ and $f(x)=4 x-1$ then find $g(f(x))$ A) $2 x-1$ B) $2 x$ C) $16 x^{2}+1$ D) $\frac{x-1}{4}$ E) None of these $\square$

Solution

Solution Steps

To find $ g(f(x)) $, we need to substitute the expression for $ f(x) $ into the function $ g(x) $. First, calculate $ f(x) = 4x - 1 $. Then, substitute this expression into $ g(x) = \frac{x-1}{2} $ by replacing $ x $ with $ f(x) $. Simplify the resulting expression to find the correct answer.

Step 1: Define the Functions

Given the functions $ g(x) = \frac{x-1}{2} $ and $ f(x) = 4x - 1 $, we need to find the composition $ g(f(x)) $.

Step 2: Substitute $ f(x) $ into $ g(x) $

Substitute $ f(x) = 4x - 1 $ into $ g(x) $: \[ g(f(x)) = g(4x - 1) = \frac{(4x - 1) - 1}{2} \]

Step 3: Simplify the Expression

Simplify the expression: \[ g(f(x)) = \frac{4x - 2}{2} = \frac{4x}{2} - \frac{2}{2} = 2x - 1 \]