Question 1165124: g={(-7,1)(-4,5)(1,-1)(7,3)}
h(x)=4x-9
Find the following.
g^-1(1)=
h^-1(x)=
(h^-1∘h)(1)=
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! Here are the solutions for the inverse function problems.
## 1. Find $g^{-1}(1)$
The function $g$ is defined by a set of ordered pairs:
$$g = \{(-7, 1), (-4, 5), (1, -1), (7, 3)\}$$
The inverse function, $g^{-1}$, reverses the ordered pairs: if $(a, b)$ is in $g$, then $(b, a)$ is in $g^{-1}$.
$$g^{-1} = \{(1, -7), (5, -4), (-1, 1), (3, 7)\}$$
To find $g^{-1}(1)$, look for the pair in $g^{-1}$ where the input (x-value) is 1.
The pair is $(1, -7)$.
$$\mathbf{g^{-1}(1) = -7}$$
*Alternatively, we look for the output (y-value) of 1 in the original function $g$. Since $g(-7) = 1$, then $g^{-1}(1) = -7$.*
## 2. Find $h^{-1}(x)$
The function $h(x)$ is given by:
$$h(x) = 4x - 9$$
To find the inverse function, $h^{-1}(x)$:
1. **Replace $h(x)$ with $y$**:
$$y = 4x - 9$$
2. **Swap $x$ and $y$**:
$$x = 4y - 9$$
3. **Solve for $y$**:
$$x + 9 = 4y$$
$$y = \frac{x + 9}{4}$$
4. **Replace $y$ with $h^{-1}(x)$**:
$$\mathbf{h^{-1}(x) = \frac{x + 9}{4}}$$
## 3. Find $(h^{-1} \circ h)(1)$
The composition $(h^{-1} \circ h)(x)$ is defined as $h^{-1}(h(x))$.
Since $h(x)$ and $h^{-1}(x)$ are inverses of each other, their composition always returns the original input, $x$, for any value in the domain.
$$(h^{-1} \circ h)(x) = x$$
Therefore, for the input $x=1$:
$$(h^{-1} \circ h)(1) = 1$$
*Alternatively, solving step-by-step:*
1. **Find $h(1)$**:
$$h(1) = 4(1) - 9 = 4 - 9 = -5$$
2. **Find $h^{-1}(h(1))$, which is $h^{-1}(-5)$**:
$$h^{-1}(-5) = \frac{(-5) + 9}{4} = \frac{4}{4} = 1$$
$$\mathbf{(h^{-1} \circ h)(1) = 1}$$
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