I need help with this problem (composite functions):
If f(x) = 4x-3 and h(x )= 4x^2-21, find a function g such that f(g(x)) = h.
I don't understand how we're supposed to combine the functions.
Oh, and the answer is supposed to be g(x) = x^2 - (9/2).
Thanks!
I was reading, and got the answer in my head before I read your answer line.
It is easy.
f(x) is given. 4x-3 YOu want 4x^2-21
so what times 4 will equal 4x^2? Ans x^2
Now what times 4 will equal -21 when you add -3? 4z-3=-21
4z=-18
z=-9/2
so g(x)=x^2 -9/2
4 x 3 = 12
21 – 2 = 19
4 x 2 = 8
5 x 4 = 20
2 + 1 = 3
14 x 12 = 168
4 x 12 = 48
2 + 4 = 6
To find the function g such that f(g(x)) equals h(x), we need to substitute g(x) into the function f(x) and set it equal to h(x).
Given that f(x) = 4x - 3, we let g(x) = y (a new variable) and substitute it into f(x):
f(g(x)) = 4g(x) - 3
Now, we set this expression equal to h(x) and solve for g(x):
4g(x) - 3 = h(x)
Since h(x) = 4x^2 - 21, we get:
4g(x) - 3 = 4x^2 - 21
To isolate g(x), we add 3 to both sides of the equation:
4g(x) = 4x^2 - 18
Next, we divide both sides by 4 to solve for g(x):
g(x) = (4x^2 - 18) / 4
Simplifying further:
g(x) = x^2 - 9/2
Therefore, g(x) = x^2 - 9/2, which matches the given answer.
To find the function g such that f(g(x)) = h, we need to substitute g(x) into the function f(x) and then set it equal to h(x).
Let's start by substituting g(x) into f(x):
f(g(x)) = 4(g(x)) - 3
Now, since we want this expression to be equal to h(x), we can set it equal to h(x):
4(g(x)) - 3 = h(x)
Next, we substitute h(x) into the equation:
4(g(x)) - 3 = 4x^2-21
Now, we can solve this equation for g(x):
4(g(x)) = 4x^2-21+3
4(g(x)) = 4x^2-18
Dividing by 4, we get:
g(x) = (4x^2-18) / 4
Simplifying this expression further:
g(x) = x^2 - (18/4)
g(x) = x^2 - (9/2)
Therefore, the function g(x) that satisfies f(g(x)) = h(x) is g(x) = x^2 - (9/2).