Questions: Question 8 The one-to-one functions g and h are defined as follows. g=(-8,8),(2,-8),(4,6),(6,-5) h(x)=2x-3 Find the following. g^-1(-8)= h^-1(x)= (h^-1 cdot h)(3)=

Solution Steps

The function \( g \) is given as a set of ordered pairs: \[ g = \{(-8,8),(2,-8),(4,6),(6,-5)\} \]

To find \( g^{-1}(-8) \), we need to find the \( x \) value such that \( g(x) = -8 \). From the set of ordered pairs, we see that: \[ g(2) = -8 \]

Thus, \( g^{-1}(-8) = 2 \).

The function \( h(x) \) is given by: \[ h(x) = 2x - 3 \]

To find the inverse function \( h^{-1}(x) \), we solve for \( x \) in terms of \( y \) where \( y = h(x) \): \[ y = 2x - 3 \]

Add 3 to both sides: \[ y + 3 = 2x \]

Divide by 2: \[ x = \frac{y + 3}{2} \]

Thus, the inverse function is: \[ h^{-1}(x) = \frac{x + 3}{2} \]

First, find \( h(3) \): \[ h(3) = 2(3) - 3 = 6 - 3 = 3 \]

Now, evaluate \( h^{-1}(3) \) using the inverse function found in Step 2: \[ h^{-1}(3) = \frac{3 + 3}{2} = \frac{6}{2} = 3 \]

Thus, \( \left(h^{-1} \cdot h\right)(3) = 3 \).

Final Answer