Question 1165630: g={(-4,-9),(-2,8),(4,5),(8,4)}
h(x)=2x-13
Find the following
g^-1(8)=
h^-1(x)=
(h∘h^-1)(-5)=
Found 3 solutions by CPhill, greenestamps, math_tutor2020:
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! This problem involves finding inverse functions and evaluating a function composition.
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## 1. Finding $g^{-1}(8)$
The function $g$ is defined by the set of ordered pairs:
$$g = \{(-4,-9), (-2,8), (4,5), (8,4)\}$$
The inverse function $g^{-1}$ reverses the ordered pairs. To find $g^{-1}(8)$, we look for the pair in $g$ where the output (y-value) is 8.
The pair in $g$ with an output of 8 is **$(-2, 8)$**.
Therefore, the input (x-value) for $g^{-1}(8)$ is $-2$.
$$\mathbf{g^{-1}(8) = -2}$$
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## 2. Finding $h^{-1}(x)$
The function is $h(x) = 2x - 13$. To find the inverse, we set $y = h(x)$ and solve for $x$ in terms of $y$.
1. Set $y = h(x)$:
$$y = 2x - 13$$
2. Swap $x$ and $y$:
$$x = 2y - 13$$
3. Solve for $y$:
$$x + 13 = 2y$$
$$y = \frac{x + 13}{2}$$
$$\mathbf{h^{-1}(x) = \frac{x + 13}{2}}$$
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## 3. Finding $(h \circ h^{-1})(-5)$
The expression $(h \circ h^{-1})(-5)$ represents the composition of the function $h$ with its inverse $h^{-1}$, evaluated at $x=-5$.
By the definition of an inverse function, when a function is composed with its inverse, the result is the identity function:
$$(f \circ f^{-1})(x) = x$$
Therefore, for $h(x)$:
$$(h \circ h^{-1})(-5) = -5$$
$$\mathbf{(h \circ h^{-1})(-5) = -5}$$
Answer by greenestamps(13258)  (Show Source):
You can put this solution on YOUR website!
g={(-4,-9),(-2,8),(4,5),(8,4)}
h(x)=2x-13
(1) Find g^(-1)(8)
Here we are asked to find "g inverse of 8". By the definition of inverse functions, that is the input value that produces an output value of 8.
By inspection of the definition of the function g, the answer is -2.
ANSWER: -2
(2) Find h^(-1)(x)
Here we are asked to find the inverse of the given function h. The standard process for finding the inverse of a function is as shown in the other response to your post: switch the x and y and solve for the new y.
For simple functions like the given h(x), an easier and faster way is to use the concept that an inverse function "gets you back where you started". To get back where you started, you need to "retrace your steps", which means perform the opposite operations in the opposite order.
In this example, the given function performs the following operations: multiply by 2; add 13. So the inverse function must perform the following operations: subtract 13; divide by 2.
ANSWER: (x-13)/2
(3) Find (h∘h^-1)(-5)
Again, by definition, an inverse function "gets you back where you started". So operating on a given input by the inverse of a function and then operating on the result by the original function gets you back where you started.
ANSWER: (h∘h^-1)(-5) = -5
Answer by math_tutor2020(3828)  (Show Source):
You can put this solution on YOUR website!
I'll focus on the second question only.
Let's say that j(x) is the inverse of h(x).
If h and j were inverses of each other, then these two equations must be true:
h( j(x) ) = x
j( h(x) ) = x
Let's use the first equation to say the following:
h(x) = 2x-13
h( j(x) ) = 2*j(x)-13 .... replace every x with j(x)
x = 2*j(x)-13
x+13 = 2*j(x)
j(x) = (x+13)/2 is the inverse of h(x)
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