dentify the inverse to the function f(x)=3x/4+1/2

Question

.(1 point)
Responses

f^−1(x)=4x/3−1/2
f inverse of x is equal to 4 x over 3 minus 1 half

f^−1(x)=4x−2/3
f inverse of x is equal to the fraction with numerator 4 x minus 2 and denominator 3

f/−1(x)=3x/4−1/2
f inverse of x is equal to 3 x over 4 minus 1 half

Answer 1

To find the inverse of the function \( f(x) = \frac{3x}{4} + \frac{1}{2} \), we will follow these steps:

1. Replace \( f(x) \) with \( y \): \[ y = \frac{3x}{4} + \frac{1}{2} \]

2. Swap \( x \) and \( y \): \[ x = \frac{3y}{4} + \frac{1}{2} \]

3. Solve for \( y \):

   • Subtract \( \frac{1}{2} \) from both sides: \[ x - \frac{1}{2} = \frac{3y}{4} \]
   • Multiply both sides by 4 to eliminate the fraction: \[ 4\left(x - \frac{1}{2}\right) = 3y \] \[ 4x - 2 = 3y \]
   • Now, divide by 3: \[ y = \frac{4x - 2}{3} \]

Thus, the inverse function is: \[ f^{-1}(x) = \frac{4x - 2}{3} \]

So, the correct response is: f^−1(x)=4x−2/3
f inverse of x is equal to the fraction with numerator 4 x minus 2 and denominator 3