For all values of \( x, f(x)=2 x-3 \) and \( g(x)=x^{2}+2 \). Solve \( f g(x)=g f(x) \) by finding the values of \( x \). Select one: a. \( \quad x=0 \) b. \( \quad x=-5,-1 \) c. \( \quad x=5,1 \) d. \( \quad x=-5,1 \)

We are given \[ f(x)=2x-3 \quad \text{and} \quad g(x)=x^2+2. \] We need to solve \[ f(g(x))=g(f(x)). \] **Step 1. Compute \( f(g(x)) \):** Substitute \( g(x)=x^2+2 \) into \( f \): \[ f(g(x))=f(x^2+2)=2(x^2+2)-3=2x^2+4-3=2x^2+1. \] **Step 2. Compute \( g(f(x)) \):** Substitute \( f(x)=2x-3 \) into \( g \): \[ g(f(x))=g(2x-3)=(2x-3)^2+2. \] Expand \( (2x-3)^2 \): \[ (2x-3)^2=4x^2-12x+9. \] Thus, \[ g(f(x))=4x^2-12x+9+2=4x^2-12x+11. \] **Step 3. Set \( f(g(x)) \) equal to \( g(f(x)) \):** \[ 2x^2+1=4x^2-12x+11. \] **Step 4. Solve the equation:** Subtract \( 2x^2+1 \) from both sides: \[ 0=4x^2-12x+11-2x^2-1, \] which simplifies to: \[ 0=2x^2-12x+10. \] Divide both sides by 2: \[ 0=x^2-6x+5. \] Now factor the quadratic: \[ x^2-6x+5=(x-5)(x-1)=0. \] Set each factor to zero: \[ x-5=0 \quad \text{or} \quad x-1=0, \] giving the solutions: \[ x=5 \quad \text{or} \quad x=1. \] Thus, the values of \( x \) that satisfy \( f(g(x))=g(f(x)) \) are \( x=5 \) and \( x=1 \). The correct answer is: c. \( \quad x=5,1 \)