Questions: Given that f(x)=x^2-7 x and g(x)=x-3, calculate (a) (f ∘ g)(x)= □ (b) (g ∘ f)(x)= □ (c) (f ∘ f)(x)= □ (d) (g ∘ g)(x)= □

Transcript text: Given that $f(x)=x^{2}-7 x$ and $g(x)=x-3$, calculate (a) $(f \circ g)(x)=$ $\square$ (b) $(g \circ f)(x)=$ $\square$ (c) $(f \circ f)(x)=$ $\square$ (d) $(g \circ g)(x)=$ $\square$

Solution

Solution Steps

Solution Approach

To solve the given problems, we need to perform function composition.

(a) For $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$.

(b) For $(g \circ f)(x)$, substitute $f(x)$ into $g(x)$.

Step 1: Understand the Problem

We are given two functions:

$ f(x) = x^2 - 7x $
$ g(x) = x - 3 $

We need to calculate the compositions:

$ (f \circ g)(x) $
$ (g \circ f)(x) $
$ (f \circ f)(x) $

Step 2: Calculate $ (f \circ g)(x) $

The composition $ (f \circ g)(x) $ means we substitute $ g(x) $ into $ f(x) $.

Substitute $ g(x) = x - 3 $ into $ f(x) $: \[ f(g(x)) = f(x - 3) = (x - 3)^2 - 7(x - 3) \]
Expand $ (x - 3)^2 $: \[ (x - 3)^2 = x^2 - 6x + 9 \]
Substitute back into the expression: \[ f(x - 3) = x^2 - 6x + 9 - 7(x - 3) \]
Distribute the $-7$: \[ -7(x - 3) = -7x + 21 \]
Combine like terms: \[ f(x - 3) = x^2 - 6x + 9 - 7x + 21 = x^2 - 13x + 30 \]

Step 3: Calculate $ (g \circ f)(x) $

The composition $ (g \circ f)(x) $ means we substitute $ f(x) $ into $ g(x) $.

Substitute $ f(x) = x^2 - 7x $ into $ g(x) $: \[ g(f(x)) = g(x^2 - 7x) = (x^2 - 7x) - 3 \]
Simplify the expression: \[ g(f(x)) = x^2 - 7x - 3 \]

Step 4: Calculate $ (f \circ f)(x) $

The composition $ (f \circ f)(x) $ means we substitute $ f(x) $ into itself.

Substitute $ f(x) = x^2 - 7x $ into $ f(x) $: \[ f(f(x)) = f(x^2 - 7x) = (x^2 - 7x)^2 - 7(x^2 - 7x) \]
Expand $ (x^2 - 7x)^2 $: \[ (x^2 - 7x)^2 = x^4 - 14x^3 + 49x^2 \]
Distribute the $-7$: \[ -7(x^2 - 7x) = -7x^2 + 49x \]
Combine like terms: \[ f(f(x)) = x^4 - 14x^3 + 49x^2 - 7x^2 + 49x = x^4 - 14x^3 + 42x^2 + 49x \]