one to one functions g and h are defined as
g={(-4,5), (2,-5),(3,2, (4,0)}
h(x)= 2x+9
find the following
g^-1 (2)
h^-1 x)
(h*h^1)(-7)
g^-1 = { (5,-4), (2,3), (0,4)}
h^-1(x) = (x-9)/2
g^-1 (2) = 3
h^-1 (x) = (x-9)/2
(h*h^1) (-7) .... is there a typo?
did you mean (h*h^-1) (-7) ??
if so, then (h*h^-1) (x)
= (2x+9)(x-9)/2
= (2x^2 + 9x - 81)/2
(h*h^-1) (-7) = (98 - 63 - 81)/2 = -23
To find the outputs for the given functions and their inverses, you can follow these steps:
1. Find g^-1(2), the inverse of function g, at x = 2.
- To find the inverse, simply swap the x and y values of the ordered pairs in function g.
In other words, interchange the first and second elements of each pair.
- g^-1 = {(5,-4), (-5,2), (2,3), (0,4)}
- Now, look for the pair where x = 2. In this case, (2,3) is the correct one.
- Therefore, g^-1(2) = 3.
2. Find h^-1(x), the inverse of function h, at x = 7.
- To find the inverse, replace h(x) with y and solve for x.
- Start with y = 2x + 9 and replace y with x and x with y.
- x = 2y + 9
- Rearrange the equation to solve for y: 2y = x - 9, y = (x - 9)/2
- Therefore, h^-1(x) = (x - 9)/2.
3. Find (h * h^-1)(-7).
- (h * h^-1)(-7) is the composition of functions h and h^-1 at x = -7.
- First, evaluate h^-1(-7) using the inverse function h^-1(x) = (x - 9)/2:
h^-1(-7) = (-7 - 9)/2 = -16/2 = -8.
- Now, substitute the value found into h(x): h(-8) = 2(-8) + 9 = -16 + 9 = -7.
So, the answers to the given questions are:
1. g^-1(2) = 3.
2. h^-1(x) = (x - 9)/2.
3. (h * h^-1)(-7) = -7.
To find the inverse of a function g, we switch the x and y values of each point in g and write it as a set.
Step 1:
Given g = {(-4,5), (2,-5), (3,2), (4,0)}
Switching x and y values, we obtain the inverse of g:
g^(-1) = {(5,-4), (-5,2), (2,3), (0,4)}
Therefore, g^(-1) = {(5,-4), (-5,2), (2,3), (0,4)}.
Step 2:
To find the inverse of the function h(x) = 2x + 9, we replace h(x) with y and solve for x.
y = 2x + 9
Switching x and y, we get:
x = 2y + 9
Now, solve for y:
x - 9 = 2y
y = (x - 9) / 2
Therefore, h^(-1) = (x - 9) / 2.
Step 3:
To find (h * h^(-1))(-7), first substitute h^(-1) into h:
h^(-1) = (x - 9) / 2
h((x - 9) / 2) = 2((x - 9) / 2) + 9
Simplifying:
h^(-1) = (x - 9) / 2
h((x - 9) / 2) = x - 9 + 9
h((x - 9) / 2) = x
Therefore, (h * h^(-1))(-7) = -7.