Find (a) \( f(g(x)) \), (b) \( g(f(x)) \), and (c) \( f(f(x)) \). \( f(x)=-x+11, g(x)=\sqrt[3]{x-3} \) a. \( f(g(x))=\square \) b. \( g(f(x))=\square \) c. \( f(f(x))=\square \)

Evaluate the expression by following steps: - step0: Evaluate: \(-x+11\) - step1: Substitute: \(-\sqrt[3]{x-3}+11\) Substitute \( x=-x+11 \) into the expression \( -x+11 \). Evaluate the expression by following steps: - step0: Evaluate: \(-x+11\) - step1: Substitute: \(-\left(-x+11\right)+11\) - step2: Multiply the terms: \(x-11+11\) - step3: Remove the opposites: \(x\) Substitute \( x=-x+11 \) into the expression \( \sqrt[3]{x-3} \). Evaluate the expression by following steps: - step0: Evaluate: \(\sqrt[3]{x-3}\) - step1: Substitute: \(\sqrt[3]{-x+11-3}\) - step2: Subtract the numbers: \(\sqrt[3]{-x+8}\) Let's find the compositions step by step: ### Given Functions: - \( f(x) = -x + 11 \) - \( g(x) = \sqrt[3]{x - 3} \) ### (a) Finding \( f(g(x)) \): To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\sqrt[3]{x - 3}\right) = -\sqrt[3]{x - 3} + 11 \] Thus, \[ f(g(x)) = -\sqrt[3]{x - 3} + 11 \] ### (b) Finding \( g(f(x)) \): To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(-x + 11) = \sqrt[3]{(-x + 11) - 3} = \sqrt[3]{-x + 8} \] Thus, \[ g(f(x)) = \sqrt[3]{-x + 8} \] ### (c) Finding \( f(f(x)) \): To find \( f(f(x)) \), we substitute \( f(x) \) into itself: \[ f(f(x)) = f(-x + 11) = -(-x + 11) + 11 = x \] Thus, \[ f(f(x)) = x \] ### Final Results: - a. \( f(g(x)) = -\sqrt[3]{x - 3} + 11 \) - b. \( g(f(x)) = \sqrt[3]{-x + 8} \) - c. \( f(f(x)) = x \)