Final Answer:
The function compositions f∘g and g∘f are as follows:
f∘g(x) = 5(3x² + 4) - 1 = 15x² + 20 - 1 = 15x² + 19
g∘f(x) = 3(5x - 1)² + 4 = 3(25x² - 10x + 1) + 4 = 75x² - 30x + 3 + 4 = 75x² - 30x + 7
Step-by-step explanation:
To find the compositions f∘g and g∘f, we start by understanding the concept of function composition, denoted as f∘g and g∘f. The notation f∘g(x) means applying function f to the result of function g for the input x. Similarly, g∘f(x) means applying function g to the result of function f for the input x.
For f∘g(x), we substitute g(x) into f(x), which gives us f(g(x)). Therefore, f∘g(x) = f(g(x)) = f(3x² + 4). Substituting f(x) = 5x - 1 into g(x), we get f∘g(x) = 5(3x² + 4) - 1 = 15x² + 20 - 1 = 15x² + 19.
For g∘f(x), we substitute f(x) into g(x), which gives us g(f(x)). Thus, g∘f(x) = g(f(x)) = g(5x - 1). Substituting g(x) = 3x² + 4 into f(x), we get g∘f(x) = 3(5x - 1)² + 4 = 3(25x² - 10x + 1) + 4 = 75x² - 30x + 3 + 4 = 75x² - 30x + 7.
Therefore, after substituting the given functions into each other successively for f∘g and g∘f, the resulting functions are f∘g(x) = 15x² + 19 and g∘f(x) = 75x² - 30x + 7.