The one-to-one functions and are defined as follows.
g={(-3,6),(-2,4),(3,7),(6,-9)}
h(x)=2x-9
find the following
g^-1(6)=
h^-1(x)=
(h^-1 oh)(1)=
To find \( g^{-1}(6) \), we need to find the input value in \( g \) that corresponds to the output value of 6.
Here's how we can find it:
1. Look for the ordered pair in the function \( g \) where the second component (or the output value) is 6. In this case, the ordered pair is \((-3, 6)\).
2. The first component of this ordered pair (-3) is the input value that corresponds to the output value 6.
3. Therefore, \( g^{-1}(6) = -3 \).
Now, let's find \( h^{-1}(x) \):
The function \( h \) is defined as \( h(x) = 2x - 9 \), and we want to find its inverse.
Here's how we can find it:
1. Start with the expression \( y = h(x) = 2x - 9 \).
2. Swap the roles of \( x \) and \( y \) (replace \( y \) with \( x \) and \( x \) with \( y \)): \( x = 2y - 9 \).
3. Solve this equation for \( y \) to find \( h^{-1}(x) \):
- Add 9 to both sides of the equation: \( x + 9 = 2y \).
- Divide both sides by 2: \( \frac{{x + 9}}{2} = y \), or equivalently, \( y = \frac{{x + 9}}{2} \).
4. Therefore, \( h^{-1}(x) = \frac{{x + 9}}{2} \).
Finally, let's find \( (h^{-1} \circ h)(1) \):
To find the composition \( h^{-1} \circ h \), we need to follow these steps:
1. Substitute the function \( h \) into the inverse function \( h^{-1} \):
\( (h^{-1} \circ h)(x) = h^{-1}(h(x)) \).
2. Substitute \( 1 \) for \( x \):
\( (h^{-1} \circ h)(1) = h^{-1}(h(1)) \).
3. Calculate \( h(1) \):
\( h(1) = 2(1) - 9 = -7 \).
4. Substitute this result into the inverse function \( h^{-1} \):
\( (h^{-1} \circ h)(1) = h^{-1}(-7) \).
5. Use the inverse function \( h^{-1}(x) = \frac{{x + 9}}{2} \) to find the output:
\( (h^{-1} \circ h)(1) = \frac{{-7 + 9}}{2} = 1 \).
Therefore, \( (h^{-1} \circ h)(1) = 1 \).