3. Let \( p(x) \) be the function that results from applying three transformations to the graph of \( f \) in this order: a horizontal dilation by a factor of 3 , a reflection over the \( x \) axis, and a vertical translation by -4 units. \( \begin{array}{lll}\text { (a) Find } p(3) & \text { (b) Find } p(-6) & \text { (c) If } p(x)=f(x) \text {, find } x \text {. }\end{array} \)

To find the function \( p(x) \) that results from applying the specified transformations to the function \( f(x) \), we will go through each transformation step by step. 1. **Horizontal Dilation by a factor of 3**: This transformation modifies the input of the function. The transformation can be represented as: \[ g(x) = f\left(\frac{x}{3}\right) \] 2. **Reflection over the \( x \)-axis**: This transformation changes the output of the function. The transformation can be represented as: \[ h(x) = -g(x) = -f\left(\frac{x}{3}\right) \] 3. **Vertical Translation by -4 units**: This transformation shifts the entire graph down by 4 units. The transformation can be represented as: \[ p(x) = h(x) - 4 = -f\left(\frac{x}{3}\right) - 4 \] Thus, the function \( p(x) \) is given by: \[ p(x) = -f\left(\frac{x}{3}\right) - 4 \] Now, we can use this function to answer the questions: ### (a) Find \( p(3) \) Substituting \( x = 3 \) into \( p(x) \): \[ p(3) = -f\left(\frac{3}{3}\right) - 4 = -f(1) - 4 \] ### (b) Find \( p(-6) \) Substituting \( x = -6 \) into \( p(x) \): \[ p(-6) = -f\left(\frac{-6}{3}\right) - 4 = -f(-2) - 4 \] ### (c) If \( p(x) = f(x) \), find \( x \) Setting \( p(x) = f(x) \): \[ -f\left(\frac{x}{3}\right) - 4 = f(x) \] Rearranging gives: \[ -f\left(\frac{x}{3}\right) = f(x) + 4 \] Multiplying both sides by -1: \[ f\left(\frac{x}{3}\right) = -f(x) - 4 \] To find \( x \), we would need more information about the function \( f(x) \) to solve this equation. Without knowing the specific form of \( f(x) \), we cannot determine the exact value of \( x \). In summary: - \( p(3) = -f(1) - 4 \) - \( p(-6) = -f(-2) - 4 \) - The equation \( f\left(\frac{x}{3}\right) = -f(x) - 4 \) needs more information about \( f(x) \) to solve for \( x \).