Answer :
To solve the problem of finding [tex]\(f(f^{-1}(14))\)[/tex] for the function [tex]\(f(x) = 3x + 2\)[/tex], follow these steps:
1. Determine the Inverse Function:
- Start with the equation [tex]\(y = 3x + 2\)[/tex].
- To find the inverse, solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
y = 3x + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
y - 2 = 3x
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{y - 2}{3}
\][/tex]
- Therefore, the inverse function is [tex]\(f^{-1}(y) = \frac{y - 2}{3}\)[/tex].
2. Calculate [tex]\(f^{-1}(14)\)[/tex]:
- Substitute [tex]\(14\)[/tex] for [tex]\(y\)[/tex] in the inverse function:
[tex]\[
f^{-1}(14) = \frac{14 - 2}{3} = \frac{12}{3} = 4
\][/tex]
3. Find [tex]\(f(f^{-1}(14))\)[/tex]:
- We now need to calculate [tex]\(f(4)\)[/tex] using the original function [tex]\(f(x) = 3x + 2\)[/tex]:
[tex]\[
f(4) = 3 \times 4 + 2 = 12 + 2 = 14
\][/tex]
So, the value of [tex]\(f(f^{-1}(14))\)[/tex] is [tex]\(14\)[/tex].
1. Determine the Inverse Function:
- Start with the equation [tex]\(y = 3x + 2\)[/tex].
- To find the inverse, solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
y = 3x + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
y - 2 = 3x
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{y - 2}{3}
\][/tex]
- Therefore, the inverse function is [tex]\(f^{-1}(y) = \frac{y - 2}{3}\)[/tex].
2. Calculate [tex]\(f^{-1}(14)\)[/tex]:
- Substitute [tex]\(14\)[/tex] for [tex]\(y\)[/tex] in the inverse function:
[tex]\[
f^{-1}(14) = \frac{14 - 2}{3} = \frac{12}{3} = 4
\][/tex]
3. Find [tex]\(f(f^{-1}(14))\)[/tex]:
- We now need to calculate [tex]\(f(4)\)[/tex] using the original function [tex]\(f(x) = 3x + 2\)[/tex]:
[tex]\[
f(4) = 3 \times 4 + 2 = 12 + 2 = 14
\][/tex]
So, the value of [tex]\(f(f^{-1}(14))\)[/tex] is [tex]\(14\)[/tex].
