Questions: f(x)=x^2-5x, g(x)=3x+1, and h(x)=(x-1)/x a) (g ∘ f)(x) b) (h ∘ g)(x) c) (f ∘ g)(x)

Solution Steps

To solve the given problems, we need to find the composition of functions. Specifically, we need to: a) Find \( (g \circ f)(x) \), which means we need to substitute \( f(x) \) into \( g(x) \). b) Find \( (h \circ g)(x) \), which means we need to substitute \( g(x) \) into \( h(x) \). c) Find \( (f \circ g)(x) \), which means we need to substitute \( g(x) \) into \( f(x) \).

To find \( (g \circ f)(2) \), we first compute \( f(2) \): \[ f(2) = 2^2 - 5 \cdot 2 = 4 - 10 = -6 \] Next, we substitute \( f(2) \) into \( g(x) \): \[ g(f(2)) = g(-6) = 3 \cdot (-6) + 1 = -18 + 1 = -17 \] Thus, \( (g \circ f)(2) = -17 \).

To find \( (h \circ g)(2) \), we first compute \( g(2) \): \[ g(2) = 3 \cdot 2 + 1 = 6 + 1 = 7 \] Next, we substitute \( g(2) \) into \( h(x) \): \[ h(g(2)) = h(7) = \frac{7 - 1}{7} = \frac{6}{7} \approx 0.8571 \] Thus, \( (h \circ g)(2) \approx 0.8571 \).

To find \( (f \circ g)(2) \), we first compute \( g(2) \) again: \[ g(2) = 7 \] Next, we substitute \( g(2) \) into \( f(x) \): \[ f(g(2)) = f(7) = 7^2 - 5 \cdot 7 = 49 - 35 = 14 \] Thus, \( (f \circ g)(2) = 14 \).

Final Answer