Final answer:
To define (g∘f)∘g, we first find g∘f by substituting f(x) into g(x), resulting in the function 2x² - 7. Then, we substitute g(x) into this new function and simplify, ultimately reaching the composed function 8x² + 24x + 11.
Step-by-step explanation:
To define (g∘f)∘g, we first need to understand what the composition of functions means. The composition of two functions is a new function that results from applying one function to the result of another. Here, g∘f means we apply f first, then apply g to the result.
Let's start with the innermost composition first, which is g∘f. Given f(x) = x² − 5 and g(x) = 3 + 2x, we substitute f(x) into g(x):
g(f(x)) = g(x² − 5) = 3 + 2(x² − 5) = 3 + 2x² − 10 = 2x² − 7
Now, we have a new function g∘f which is 2x² − 7. To find (g∘f)∘g, we now need to substitute g(x) into this new function:
(g∘f)(g(x)) = 2g(x)² − 7 = 2(3 + 2x)² − 7
First, we expand the square:
(3 + 2x)² = 9 + 12x + 4x²
Then, multiply this expression by 2 and subtract 7:
2(9 + 12x + 4x²) − 7 = 18 + 24x + 8x² − 7 = 8x² + 24x + 11
The composed function (g∘f)∘g is 8x² + 24x + 11.