Certainly! Let's go through a step-by-step solution to find the expression for [tex]\( f(g(x)) \)[/tex].
Given functions:
[tex]\[ f(x) = x^2 - 10 \][/tex]
[tex]\[ g(x) = 5x + 6 \][/tex]
[tex]\[ h(x) = \frac{1}{2}x + 8 \][/tex]
[tex]\[ j(x) = \sqrt{x + 3} \][/tex]
Since we need to find [tex]\( f(g(x)) \)[/tex], we will focus on the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
1. Determine the expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 5x + 6 \][/tex]
2. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
To find [tex]\( f(g(x)) \)[/tex], we substitute [tex]\( g(x) \)[/tex] into the function [tex]\( f(x) \)[/tex]. This means replacing every instance of [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex].
We know that:
[tex]\[ f(x) = x^2 - 10 \][/tex]
So,
[tex]\[ f(g(x)) = f(5x + 6) \][/tex]
3. Evaluate [tex]\( f(5x + 6) \)[/tex]:
Substitute [tex]\( 5x + 6 \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[ f(5x + 6) = (5x + 6)^2 - 10 \][/tex]
Therefore, the expression for [tex]\( f(g(x)) \)[/tex] is:
[tex]\[ f(g(x)) = (5x + 6)^2 - 10 \][/tex]
This is the detailed step-by-step process for finding the expression [tex]\( f(g(x)) \)[/tex].
