F(x)= -x+2 and g(x)=(x-3)^2 an algebraic model for
A) y=f(g(x))
B) y=f(f(x))
C)y=f^-1(f(x))
A) To find the algebraic model for y = f(g(x)), we substitute g(x) into f(x) and simplify:
f(g(x)) = -g(x) + 2
= -(x-3)^2 + 2
= -(x^2 - 6x + 9) + 2
= -x^2 + 6x - 7
Therefore, the algebraic model for y = f(g(x)) is y = -x^2 + 6x - 7.
B) To find the algebraic model for y = f(f(x)), we substitute f(x) into f(x) and simplify:
f(f(x)) = -f(x) + 2
= -(-x + 2) + 2
= x - 2 + 2
= x
Therefore, the algebraic model for y = f(f(x)) is y = x.
C) To find the algebraic model for y = f^(-1)(f(x)), we first need to find the inverse function of f(x).
Given f(x) = -x + 2, to find its inverse, we solve for x:
y = -x + 2
x = -y + 2
x - 2 = -y
y = -x + 2
Therefore, the inverse function of f(x) is f^(-1)(x) = -x + 2.
Now, we substitute f(x) into f^(-1)(x) and simplify:
f^(-1)(f(x)) = -f(x) + 2
= -(-x + 2) + 2
= x - 2 + 2
= x
Therefore, the algebraic model for y = f^(-1)(f(x)) is y = x.
A) To find an algebraic model for y = f(g(x)), we substitute the expression for g(x) into f(x). So, we have:
f(g(x)) = f((x-3)^2) = -(x-3)^2 + 2
Therefore, an algebraic model for y = f(g(x)) is y = -(x-3)^2 + 2.
B) To find an algebraic model for y = f(f(x)), we substitute the expression for f(x) into itself. So, we have:
f(f(x)) = f(-x+2) = -(-x+2) + 2
Simplifying, we get:
f(f(x)) = x-2 + 2 = x
Therefore, an algebraic model for y = f(f(x)) is y = x.
C) To find an algebraic model for y = f^-1(f(x)), we need to find the inverse of f(x) first. Starting with f(x) = -x+2, we can swap x and y and solve for y:
x = -y+2 (swapping variables)
y = -x+2 (solving for y)
This gives us the inverse function of f(x) which is f^-1(x) = -x+2.
Now, we can substitute f(x) with f^-1(x) in y = f^-1(f(x)), and simplify:
y = f^-1(f(x)) = f^-1(-x+2) = -(-x+2)+2
Simplifying further:
y = x
Thus, an algebraic model for y = f^-1(f(x)) is also y = x.
To determine the algebraic models for the given cases, we need to substitute the function expressions into the respective function compositions.
A) For y = f(g(x)), we substitute g(x) into f(x):
f(g(x)) = -g(x) + 2
Substituting g(x) = (x-3)^2:
f(g(x)) = -((x-3)^2) + 2
Simplifying further:
f(g(x)) = -(x^2 - 6x + 9) + 2
f(g(x)) = -x^2 + 6x - 7
Therefore, the algebraic model for y = f(g(x)) is y = -x^2 + 6x - 7.
B) For y = f(f(x)), we substitute f(x) into f(x):
f(f(x)) = -f(x) + 2
Substituting f(x) = -x + 2:
f(f(x)) = -(-x + 2) + 2
Simplifying further:
f(f(x)) = x - 4
Therefore, the algebraic model for y = f(f(x)) is y = x - 4.
C) For y = f^(-1)(f(x)), we substitute f(x) into f^(-1)(x):
f^(-1)(f(x))
Since f(x) = -x + 2, we need to find the inverse of f(x) to proceed.
To find the inverse, we interchange x and y and solve for y.
x = -y + 2
y = 2 - x
Therefore, the inverse of f(x) is f^(-1)(x) = 2 - x.
Substituting f(x) = -x + 2 into f^(-1)(x), we get:
f^(-1)(f(x)) = 2 - (-x + 2)
Simplifying further:
f^(-1)(f(x)) = x
Therefore, the algebraic model for y = f^(-1)(f(x)) is y = x.
In summary:
A) y = f(g(x)) is given by y = -x^2 + 6x - 7.
B) y = f(f(x)) is given by y = x - 4.
C) y = f^(-1)(f(x)) is given by y = x.