Sure, here's a detailed step-by-step solution to the problem:
### (a) Find
#### (i) [tex]\( f(3) \)[/tex]
To find [tex]\( f(3) \)[/tex], we substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) = 3x + 5 \)[/tex].
[tex]\[ f(3) = 3(3) + 5 = 9 + 5 = 14 \][/tex]
So, [tex]\( f(3) = 14 \)[/tex].
#### (ii) [tex]\( g(x-3) \)[/tex]
To find [tex]\( g(x-3) \)[/tex], we substitute [tex]\( x-3 \)[/tex] into the function [tex]\( g(x) = 7 - 2x \)[/tex].
[tex]\[ g(x-3) = 7 - 2(x-3) = 7 - 2x + 6 = 13 - 2x \][/tex]
So, [tex]\( g(x-3) = 13 - 2x \)[/tex].
### (b) Find the inverse function [tex]\( g^{-1}(x) \)[/tex]
To find the inverse function of [tex]\( g(x) = 7 - 2x \)[/tex], follow these steps:
1. Replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]: [tex]\( y = 7 - 2x \)[/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 7 - 2x \\
2x = 7 - y \\
x = \frac{7 - y}{2}
\][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[
g^{-1}(x) = \frac{7 - x}{2}
\][/tex]
So, the inverse function [tex]\( g^{-1}(x) = \frac{7 - x}{2} \)[/tex].
### (c) Find [tex]\( h(f(x)) \)[/tex] in the simplest form
To find [tex]\( h(f(x)) \)[/tex], substitute [tex]\( f(x) = 3x + 5 \)[/tex] into the function [tex]\( h(x) = x^2 - 8 \)[/tex].
[tex]\[ h(f(x)) = h(3x + 5) = (3x + 5)^2 - 8 \][/tex]
Expand [tex]\( (3x + 5)^2 \)[/tex]:
[tex]\[ (3x + 5)^2 = 9x^2 + 30x + 25 \][/tex]
So,
[tex]\[ h(f(x)) = 9x^2 + 30x + 25 - 8 = 9x^2 + 30x + 17 \][/tex]
Thus, [tex]\( h(f(x)) = 9x^2 + 30x + 17 \)[/tex].
