Certainly! Let's go through the steps for each part of the question.
### Part (a)
We are given the function [tex]\( f(x) = 7x - 6 \)[/tex].
To find [tex]\( f(4) \)[/tex]:
1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[
f(4) = 7(4) - 6
\][/tex]
2. Perform the multiplication:
[tex]\[
7 \times 4 = 28
\][/tex]
3. Subtract 6 from 28:
[tex]\[
28 - 6 = 22
\][/tex]
Therefore, the value of [tex]\( f(4) \)[/tex] is [tex]\( 22 \)[/tex].
### Part (b)
We now need to find the value of the inverse function [tex]\( f^{-1}(22) \)[/tex].
To find the inverse function, we start by expressing [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] from the original function:
[tex]\[
y = 7x - 6
\][/tex]
Next, solve for [tex]\( x \)[/tex]:
1. Add 6 to both sides:
[tex]\[
y + 6 = 7x
\][/tex]
2. Divide both sides by 7:
[tex]\[
x = \frac{y + 6}{7}
\][/tex]
Thus, the inverse function is:
[tex]\[
f^{-1}(y) = \frac{y + 6}{7}
\][/tex]
To find [tex]\( f^{-1}(22) \)[/tex]:
1. Substitute [tex]\( y = 22 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(22) = \frac{22 + 6}{7}
\][/tex]
2. Add 22 and 6:
[tex]\[
22 + 6 = 28
\][/tex]
3. Divide 28 by 7:
[tex]\[
\frac{28}{7} = 4
\][/tex]
Therefore, the value of [tex]\( f^{-1}(22) \)[/tex] is [tex]\( 4 \)[/tex].
### Summary
- The value of [tex]\( f(4) \)[/tex] is [tex]\( 22 \)[/tex].
- The value of [tex]\( f^{-1}(22) \)[/tex] is [tex]\( 4 \)[/tex].
