Let h(x)=4-x. Use h, the table, and points on graph to evaluate the expression.
x
2 3 4 5 6
f(x)
−1 0 1 2 3
g(x) points (-1,1), (2,3), (3,5)
(a) (g^−1 compose f^ −1)(2)
(b) (g^−1 compose h)(1)
(c) (h^−1 compose f compose g^-1)(3)
To evaluate these expressions, we need to understand how to compose functions and find the inverse of a function.
(a) (g^−1 compose f^ −1)(2):
To evaluate this expression, we need to find the inverse of both f(x) and g(x). Let's start by finding the inverse of f(x) = 4 - x.
To find the inverse of f(x), we swap the x and y variables and solve for y:
x = 4 - y
Rearranging the equation, we have:
y = 4 - x
So, the inverse of f(x) is f^ −1(x) = 4 - x.
Now, let's find the inverse of g(x). To do this, we swap the x and y variables in the ordered pairs of g(x) and solve for y.
Given points for g(x):
(-1, 1), (2, 3), (3, 5)
Swapping x and y, we get the inverse points for g^−1(x):
(1, -1), (3, 2), (5, 3)
Now, let's evaluate (g^−1 compose f^ −1)(2):
(g^−1 compose f^ −1)(2) means we first apply f^ −1 to 2, and then apply g^−1 to the result.
Applying f^ −1 to 2:
f^ −1(2) = 4 - 2 = 2
Now, applying g^−1 to the result (2):
g^−1(2) = 3
Therefore, (g^−1 compose f^ −1)(2) equals 3.
(b) (g^−1 compose h)(1):
To evaluate this expression, we need to find the inverse of h(x) and the inverse of g(x). We have already found the inverse of f(x) in the previous step.
Given h(x) = 4 - x, we find the inverse by swapping x and y:
x = 4 - y
Rearranging the equation, we have:
y = 4 - x
Thus, h^ −1(x) = 4 - x.
Now, let's evaluate (g^−1 compose h)(1):
(g^−1 compose h)(1) means we first apply h to 1, and then apply g^−1 to the result.
Applying h to 1:
h(1) = 4 - 1 = 3
Now, applying g^−1 to the result (3):
g^−1(3) = 2
Hence, (g^−1 compose h)(1) equals 2.
(c) (h^−1 compose f compose g^-1)(3):
To evaluate this expression, we need to find the inverse of h(x), the inverse of f(x), and the inverse of g(x). Again, we have already found the inverses of h(x) and f(x) in the earlier steps.
Now, let's find the inverse of g(x) using the given points:
(-1, 1), (2, 3), (3, 5)
Swapping x and y, we obtain the inverse points for g^−1(x):
(1, -1), (3, 2), (5, 3)
To evaluate (h^−1 compose f compose g^-1)(3), we first apply g^-1 to 3, then apply f, and finally apply h^−1 to the result.
Applying g^-1 to 3:
g^-1(3) = 2
Next, applying f to the result (2):
f(2) = -1
Finally, applying h^−1 to the result (-1):
h^−1(-1) = 5
Therefore, (h^−1 compose f compose g^-1)(3) equals 5.