Sorry I had to repost, I left somethings out.
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
Find (f *g)(x)
I got 14x+2
For the given functions f and g, find the indicated composition.
8. f(x)=-5x+3, g(x)=6x+3
(g * f)(x)
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
If your f*g(x) means f(g(x)), then for problem #5,
f[g(x)] = 7(2x-1) + 9 = 14 x + 2
For problem #12, there may be more than one combination of f and g that result in
f[g(x)] = 1/(x^2-2). The two functions you listed work fine.
For question 5, you are correct!
For question 8, the correct answer is (g * f)(x) = -30x + 18. Good job!
For question 12, your answer is incorrect. The correct functions are f(x) = 1/x and g(x) = x^2 - 2.
And for question 14, your answer is correct! The functions are f(x) = x^5 and g(x) = -2x + 19. Well done!
To check your answers for the compositions of functions, let's go through each question step by step.
5. To find the composition (f * g)(x), you need to substitute g(x) into f(x).
Given f(x) = 7x + 9 and g(x) = 2x - 1, substitute g(x) into f(x):
(f * g)(x) = f(g(x)) = 7(2x - 1) + 9
Simplifying this expression, you get:
(f * g)(x) = 14x - 7 + 9 = 14x + 2
Hence, your answer of 14x + 2 is correct.
8. To find the composition (g * f)(x), you need to substitute f(x) into g(x).
Given f(x) = -5x + 3 and g(x) = 6x + 3, substitute f(x) into g(x):
(g * f)(x) = g(f(x)) = 6(-5x + 3) + 3
Simplifying this expression, you get:
(g * f)(x) = -30x + 18 + 3 = -30x + 21
Hence, your answer of -30x + 21 is correct.
12. To find functions f and g such that h(x) = (f * g)(x) = 1/(x^2 - 2), we need to decompose h(x) into f(x) and g(x).
Start by considering the denominator of h(x), which is (x^2 - 2).
We can decompose it as the difference of two squares: (x^2 - 2) = (x + √2)(x - √2).
Now let's assign f(x) and g(x) to each factor:
f(x) = 1/(x + √2) and g(x) = x - √2.
Therefore, your answer of f(x) = 1/x and g(x) = x^2 - 2 is incorrect.
14. To find functions f and g such that h(x) = (f * g)(x) = (-2x + 19)^5, we need to decompose h(x) into f(x) and g(x).
Since h(x) is already in the form (g(x))^5, we can assign g(x) = -2x + 19.
To find f(x), we can assign it as f(x) = x^5, which raises x to the power of 5.
Therefore, your answer of f(x) = x^5 and g(x) = -2x + 19 is correct.
Just to summarize:
- For question 5, your answer of (f * g)(x) = 14x + 2 is correct.
- For question 8, your answer of (g * f)(x) = -30x + 21 is correct.
- For question 12, your answer of f(x) = 1/x and g(x) = x^2 - 2 is incorrect.
- For question 14, your answer of f(x) = x^5 and g(x) = -2x + 19 is correct.