Question

Let f(x) = x3 6, g(x) = x2 - 8, and h(x) = 8x 7. find the rule for the function. gf

Answer 1

The rule for the composite function gf, where f(x) = x^3 and g(x) = x^2 - 8, is found by substituting f(x) into g(x), which gives us g(f(x)) = (x^3)^2 - 8, or x^6 - 8.

Step-by-step explanation:

To find the rule for the function gf, which is the composition of functions g(x) and f(x), we need to substitute f(x) into g(x). Here, f(x) = x3 and g(x) = x2 - 8. The composition (gf)(x) means we need to compute g(f(x)).

Let's perform the substitution step-by-step:

1. 
   
3. Start with g(x), which has an 'x' that needs to be replaced by f(x).
4. 
   
6. Substitute f(x) into g(x), yielding g(f(x)) = (x3)2 - 8.
7. 
   
9. Simplify the expression: g(f(x)) = x6 - 8.

Thus, the rule for the composite function gf is g(f(x)) = x6 - 8.