Summarize the pertinent information oblained by applying the graphing strategy and sketch the graph of \( y=f(\mathrm{k}) \) \[ f(x)=(x-4)\left(x^{2}-8 x-32\right) \] Select the correct choice below and, if necessary, fill in the answer box to complele your choice. A. The domain of t is \( \square \) . (Type your answet in interval nolation. Type an exacl answer, using radicals as needed. Use a corma lo separate answers as needed.) B. The domain of \( / \) is empty. Select the correct choice below and, it necessary, fill in the answer box to complelo you choice. A. The \( x \)-intercept:(s) of f is/are \( \mathrm{x}= \) \( \square \) . (Type an exact answer, using radicals as needed. Use a comma to separale answers as needed.) B. The function f has no x -intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercepl of \( f \) is \( y= \) \( \square \)

**Step 1. Determine the Domain** The function is \[ f(x) = (x-4)\left(x^2 - 8x - 32\right). \] Since it is a polynomial, its domain is all real numbers. In interval notation, the domain is \[ (-\infty,\infty). \] **Step 2. Find the \(x\)-Intercepts** To find the \(x\)-intercepts, set \(f(x)=0\): \[ (x-4)(x^2 -8x-32)=0. \] 1. The factor \(x-4=0\) gives \[ x=4. \] 2. For the quadratic \(x^2-8x-32=0\), apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}, \] where \(a=1\), \(b=-8\), and \(c=-32\). Compute the discriminant: \[ (-8)^2 - 4(1)(-32) = 64 + 128 = 192. \] Then \[ x = \frac{8 \pm \sqrt{192}}{2} = \frac{8 \pm 8\sqrt{3}}{2} = 4 \pm 4\sqrt{3}. \] Thus, the \(x\)-intercepts are at: \[ x = 4-4\sqrt{3},\quad x=4,\quad x = 4+4\sqrt{3}. \] **Step 3. Find the \(y\)-Intercept** The \(y\)-intercept is found by setting \(x=0\): \[ f(0) = (0-4)\left(0^2-8\cdot 0-32\right) = (-4)(-32) = 128. \] So, the \(y\)-intercept is \[ (0,128). \] **Step 4. Summary of Pertinent Information** - **Domain:** \((- \infty,\infty)\). - **\(x\)-Intercepts:** \(x = 4-4\sqrt{3},\quad 4,\quad 4+4\sqrt{3}.\) - **\(y\)-Intercept:** \( (0,128).\) - **End Behavior:** Since \(f(x)\) is a cubic polynomial with a positive leading coefficient, as \(x \to -\infty\), \(f(x) \to -\infty\) and as \(x \to \infty\), \(f(x) \to \infty\). **Answer Choices:** 1. **Domain:** Choice A: The domain of \(f\) is \(\boxed{(-\infty,\infty)}\). 2. **\(x\)-Intercepts:** Choice A: The \(x\)-intercepts of \(f\) are \(x = \boxed{4-4\sqrt{3},\; 4,\; 4+4\sqrt{3}}\). 3. **\(y\)-Intercept:** Choice A: The \(y\)-intercept of \(f\) is \(y = \boxed{128}\).