Questions: The one-to-one functions g and h are defined as follows g=(-9,1),(1,-2),(2,9),(6,-7) h(x)=2x+13 Find the following. g^-1(1)= h^-1(x)= (h^-1 circ h)(-5)=

Solution Steps

To solve the given problems, we need to follow these steps:

1. Finding \( g^{-1}(1) \): Since \( g \) is a one-to-one function, we can find the inverse by swapping the pairs. We need to find the \( x \) value such that \( g(x) = 1 \).

2. Finding \( h^{-1}(x) \): To find the inverse of the function \( h(x) = 2x + 13 \), we solve for \( x \) in terms of \( y \) where \( y = h(x) \).

3. Finding \( (h^{-1} \circ h)(-5) \): This involves applying the function \( h \) to \(-5\) and then applying the inverse function \( h^{-1} \) to the result.

To find \( g^{-1}(1) \), we need to determine the value of \( x \) such that \( g(x) = 1 \). From the given function \( g \):

\[ g = \{(-9,1),(1,-2),(2,9),(6,-7)\} \]

We see that \( g(-9) = 1 \). Therefore, \( g^{-1}(1) = -9 \).

The function \( h(x) \) is given by:

\[ h(x) = 2x + 13 \]

To find the inverse function \( h^{-1}(x) \), we solve for \( x \) in terms of \( y \):

\[ y = 2x + 13 \]

Rearranging to solve for \( x \):

\[ y - 13 = 2x \implies x = \frac{y - 13}{2} \]

Thus, the inverse function is:

\[ h^{-1}(x) = \frac{x - 13}{2} \]

To find \( \left(h^{-1} \circ h\right)(-5) \), we first evaluate \( h(-5) \):

\[ h(-5) = 2(-5) + 13 = -10 + 13 = 3 \]

Next, we apply \( h^{-1} \) to the result:

\[ h^{-1}(3) = \frac{3 - 13}{2} = \frac{-10}{2} = -5 \]

Thus, \( \left(h^{-1} \circ h\right)(-5) = -5 \).

Final Answer