Can you please check my answers?
For the given functions f and g, find the indicated composition.
5. f(x)=7x+9, g(x)=2x-1
I got 14x+2
For the given functions f and g, find the indicated composition.
8. f(x)=-5x+3, g(x)=6x+3
I got -30x+18
Find functions f and g so that h(x)= (f *g)(x)
12. h(x)= 1/(x^2-2)
I got (f)=1/x, g(x)=x^2-2
14. h(x)=(-2x+19)^5
I got f(x)x^5, g(x)=-2x+19
To check your answers for the composition of functions, you need to substitute the given functions into the composition formula and simplify. Let's go through each problem together:
5. The composition of f(x) and g(x) is denoted as (f ∘ g)(x). To find this, you substitute g(x) into f(x) and simplify.
(f ∘ g)(x) = f(g(x)) = f(2x - 1)
Substituting f(x) = 7x + 9 into f(2x - 1):
(f ∘ g)(x) = 7(2x - 1) + 9
= 14x - 7 + 9
= 14x + 2
So, your answer of 14x + 2 is correct!
8. Similarly, the composition of f(x) and g(x) is given by:
(f ∘ g)(x) = f(g(x)) = f(6x + 3)
Substituting f(x) = -5x + 3 into f(6x + 3):
(f ∘ g)(x) = -5(6x + 3) + 3
= -30x - 15 + 3
= -30x - 12
Your answer of -30x - 12 is correct!
12. To find functions f(x) and g(x) such that h(x) = (f ∘ g)(x) = 1/(x^2 - 2), we need to separate h(x) into the composition form.
h(x) = (f ∘ g)(x)
= f(g(x))
= f(x^2 - 2)
So the function f(x) should take the output of g(x), which is x^2 - 2, and simplify it to 1/(x^2 - 2). Therefore,
f(x) = 1/x
For g(x), we need to find the function that would produce the expression x^2 - 2 when given x as the input. Hence,
g(x) = x^2 - 2
Congratulations! Your answer of f(x) = 1/x and g(x) = x^2 - 2 is correct.
14. To find functions f(x) and g(x) such that h(x) = (f ∘ g)(x) = (-2x + 19)^5, we need to separate h(x) into the composition form.
h(x) = (f ∘ g)(x)
= f(g(x))
= f(-2x + 19)
So the function f(x) should take the output of g(x), which is -2x + 19, and simplify it to (-2x + 19)^5. Therefore,
f(x) = x^5
For g(x), we need to find the function that would produce the expression -2x + 19 when given x as the input. Hence,
g(x) = -2x + 19
Congratulations! Your answer of f(x) = x^5 and g(x) = -2x + 19 is correct.