Question

The one-to-one functions g and h are defined as follows.
g{(-9,6),(-3,1),(3,-3),(8,5)}
h(x)=2x+9

Find the following:
g^-1(-3)
h^-1(x)
(h o h^-1)(-7)

Answer 1

On inverse functions, the x and y values switch.
For the first, switch the values, for the second, switch the variables and solve for y.

Answer 2

h(0) = 2*0 + 9 = 9
h^-1 -1 = 2x + 9
x =( -1-9) / 2 = -5

Answer 3

g^-1(-3)=3
h^-1(x)=-5

Answer 4

\(f^{-1}\cdot f(x) = f^{-1}(f(x)) = x, \ \forall x\in \mathbb{R}\)

Answer 5

@mathmale can you help me with this one?

Answer 6

The one-to-one functions g is defined as follows:
g{(-9,6),(-3,1),(3,-3),(8,5)}

Find the following:
g^-1(-3)

We are asked to find the inverse function of g(x):

From g(x), we want:\[g ^{-1}(x)\]which is the symbol for "the inverse of g(x)."

We don't actually have g(x), so we can't find g^(-1)(x) algebraically.

However, there's an important and delightful principle that we need to apply here:

If f(a) = b, we have the point (a,b).
If we find the inverse function
\[f ^{-1}(x)\]
in the usual way, then the following will be true:
\[f ^{-1}(b)=a \rightarrow(b,a).\]

In other words, using a as the input to f(x) produces b;
using b as the input to \[f ^{-1}(x)\] produces a.

So... !! If we are given points generated by g(x): {(-9,6),(-3,1),(3,-3),(8,5)} , then, simply by reversing the order of the numerals within each set of parentheses, we can get the same number of points generated by
-1
g (x) : {(6,-9), (1,-3), ( you finish this, please )}

Answer 7

(-3,3),(5,8)

Answer 8

Cool! You've shown you understand what's happening here!

Answer 9

so after working on this the greater part of the night I have

g^-1(-3)= 3
h^-1(x)= -5
(h o h^-1)(-7)=-2

can anyone tell me if I did this right?

Answer 10

@ganeshie8 @whpalmer4 can either of you tell me if I did this right

Answer 11

g: {(-9,6),(-3,1),(3,-3),(8,5)} does have the inverse \[g ^{-1}(x):{(6,-9), (1,3), (-3,3), (5,8)}\]

Answer 12

Regarding h(x)=2x+9:
Let h(x)=y=2x+9, or
y=2x+9

Interchange x and y:

x=2y+9
Solve this for y: 2y + 9 = x
- 9 = -9 (subtract 9 from both sides to isolate 2y)
------ ----
2y = (x-9)
----- -----
2 2
x+9
y= -----------
2

Replace that y with h^(-1)(x) and you'll be done; you'll have the inverse of h.

Answer 13

STOP here. You're not required to evaluate that inverse at any particular x value, unless I'm sorely mistaken.

Answer 14

Regarding the last part of the hw problem:

Since h and its inverse are just that: inverse functions,

multiplying h(x) by its inverse produces simply x.

If x is 5, then the desired result is also 5.

Ask more questions about this if it's not clear. I like to see what you did, and to learn what your reasoning was, before I say "right" or "wrong".