Questions: Suppose that the functions g and h are defined as follows: g(x) = 3x^2 - 7 h(x) = 6x - 7 (a) Find (g/h)(-5). (b) Find all values that are NOT in the domain of g/h. If there is more than one value, separate them with commas. (a) (g/h)(-5) = [] (b) Value(s) that are NOT in the domain of g/h: []

Solution Steps

To solve these problems, we need to evaluate the function \((\frac{g}{h})(x)\) at a specific point and determine the domain of the function.

(a) To find \((\frac{g}{h})(-5)\), we will substitute \(x = -5\) into both \(g(x)\) and \(h(x)\), and then divide the results.

(b) The domain of \(\frac{g}{h}\) is all real numbers except where \(h(x) = 0\), since division by zero is undefined. We will solve the equation \(h(x) = 0\) to find these values.

We start by calculating the values of the functions \( g \) and \( h \) at \( x = -5 \):

\[ g(-5) = 3(-5)^2 - 7 = 3(25) - 7 = 75 - 7 = 68 \]

\[ h(-5) = 6(-5) - 7 = -30 - 7 = -37 \]

Next, we find the value of \( \left(\frac{g}{h}\right)(-5) \):

\[ \left(\frac{g}{h}\right)(-5) = \frac{g(-5)}{h(-5)} = \frac{68}{-37} \approx -1.8378 \]

To find the values that are not in the domain of \( \frac{g}{h} \), we need to solve for when \( h(x) = 0 \):

\[ h(x) = 6x - 7 = 0 \implies 6x = 7 \implies x = \frac{7}{6} \approx 1.1667 \]

Final Answer