To find [tex]\( f^{-1}(3) \)[/tex] for the function [tex]\( f(x) = 4x + 2 \)[/tex], we need to follow these steps to determine the inverse function, [tex]\( f^{-1}(x) \)[/tex], and then use it to compute the required value.
### Step 1: Define the function and set up the equation for the inverse
Given the function [tex]\( f(x) = 4x + 2 \)[/tex], we start by writing the equation that represents this function:
[tex]\[ y = 4x + 2 \][/tex]
### Step 2: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]
To find the inverse function, we need to solve this equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
1. Subtract 2 from both sides of the equation:
[tex]\[ y - 2 = 4x \][/tex]
2. Divide both sides by 4:
[tex]\[ x = \frac{y - 2}{4} \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(y) = \frac{y - 2}{4} \][/tex]
### Step 3: Compute [tex]\( f^{-1}(3) \)[/tex]
Now, we need to evaluate this inverse function at [tex]\( y = 3 \)[/tex].
1. Substitute [tex]\( y = 3 \)[/tex] into the inverse function:
[tex]\[ f^{-1}(3) = \frac{3 - 2}{4} \][/tex]
2. Simplify the expression:
[tex]\[ f^{-1}(3) = \frac{1}{4} \][/tex]
Therefore, the value of [tex]\( f^{-1}(3) \)[/tex] is:
[tex]\[ f^{-1}(3) = 0.25 \][/tex]
