Questions: Question 8 Given that f(x)=x^2-7x and g(x)=x-9, perform the indicated operation. (a) (f ∘ g)(x)= (b) (g ∘ f)(x)= (c) (f ∘ f)(x)= (d) (g ∘ g)(x)= (e) (f ∘ g)(-3)= (f) (g ∘ f)(-6)=

Transcript text: Question 8 Given that $f(x)=x^{2}-7 x$ and $g(x)=x-9$, perform the indicated operation. (a) $(f \circ g)(x)=$ $\square$ (b) $(g \circ f)(x)=$ $\square$ (c) $(f \circ f)(x)=$ $\square$ (d) $(g \circ g)(x)=$ $\square$ (e) $(f \circ g)(-3)=$ $\square$ (f) $(g \circ f)(-6)=$ $\square$

Solution

Solution Steps

Solution Approach

To solve the given problems, we need to perform function composition. This involves substituting one function into another. For each part: (a) For $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$. (b) For $(g \circ f)(x)$, substitute $f(x)$ into $g(x)$. (c) For $(f \circ f)(x)$, substitute $f(x)$ into itself. For the specific values: (e) Evaluate $(f \circ g)(x)$ at $x = -3$. (f) Evaluate $(g \circ f)(x)$ at $x = -6$.

Step 1: Define the Functions

We are given two functions:

$ f(x) = x^2 - 7x $
$ g(x) = x - 9 $

Step 2: Perform Function Composition

To find the compositions, we substitute one function into another.

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

Substitute $ g(x) $ into $ f(x) $: \[ f(g(x)) = f(x - 9) = (x - 9)^2 - 7(x - 9) \] Simplify: \[ = (x^2 - 18x + 81) - (7x - 63) = x^2 - 25x + 144 \]

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

Substitute $ f(x) $ into $ g(x) $: \[ g(f(x)) = g(x^2 - 7x) = (x^2 - 7x) - 9 = x^2 - 7x - 9 \]

Substitute $ f(x) $ into itself: \[ f(f(x)) = f(x^2 - 7x) = (x^2 - 7x)^2 - 7(x^2 - 7x) \] Simplify: \[ = (x^4 - 14x^3 + 49x^2) - (7x^2 - 49x) = x^4 - 14x^3 + 42x^2 + 49x \]

Step 3: Evaluate Specific Values

Evaluate the compositions at specific values.

(e) $(f \circ g)(-3)$

Substitute $-3$ into $(f \circ g)(x)$: \[ f(g(-3)) = f(-3 - 9) = f(-12) = (-12)^2 - 7(-12) = 144 + 84 = 228 \]

(f) $(g \circ f)(-6)$

Substitute $-6$ into $(g \circ f)(x)$: \[ g(f(-6)) = g((-6)^2 - 7(-6)) = g(36 + 42) = g(78) = 78 - 9 = 69 \]