Given that f(x) = |x| and g(x) = 4x - 7, calculate:

(a) f ∘ g(x) =
(b) g ∘ f(x) =
(c) f ∘ f(x) =
(d) g ∘ g(x) =

The compositions of the functions yield: f ∘ g(x) = |4x - 7|, g ∘ f(x) = 4|x| - 7, f ∘ f(x) = |x|, and g ∘ g(x) = 16x - 35. Each composition was calculated by substituting one function into the other. This illustrates how function operations can be performed in sequence to produce new functions.
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Answered by Anonymous | 2025-07-04

Let's calculate each of the compositions step by step:
(a) f ∘ g(x) : This means f(g(x)), which involves applying g(x) first, then f(x).
Given:

g ( x ) = 4 x − 7
f ( x ) = ∣ x ∣

To find f ( g ( x )) :

Substitute g(x) into f(x): f ( g ( x )) = f ( 4 x − 7 ) = ∣4 x − 7∣

So, f ∘ g ( x ) = ∣4 x − 7∣ .
(b) g ∘ f(x) : This means g(f(x)), which involves applying f(x) first, then g(x).
Given:

f ( x ) = ∣ x ∣
g ( x ) = 4 x − 7

To find g ( f ( x )) :

Substitute f(x) into g(x): g ( f ( x )) = g ( ∣ x ∣ ) = 4∣ x ∣ − 7

So, g ∘ f ( x ) = 4∣ x ∣ − 7 .
(c) f ∘ f(x) : This means f(f(x)), which involves applying f(x) twice.
Given:

f ( x ) = ∣ x ∣

To find f ( f ( x )) :

Substitute f(x) into itself: f ( f ( x )) = f ( ∣ x ∣ ) = ∣∣ x ∣∣

Since the absolute value of a number is always non-negative, applying the absolute value operation again does not change it. Thus: ∣∣ x ∣∣ = ∣ x ∣
So, f ∘ f ( x ) = ∣ x ∣ .
(d) g ∘ g(x) : This means g(g(x)), which involves applying g(x) twice.
Given:

g ( x ) = 4 x − 7

To find g ( g ( x )) :

Substitute g(x) into itself: g ( g ( x )) = g ( 4 x − 7 ) = 4 ( 4 x − 7 ) − 7
Simplify: = 16 x − 28 − 7 = 16 x − 35

So, g ∘ g ( x ) = 16 x − 35 .
In summary:

f ∘ g ( x ) = ∣4 x − 7∣
g ∘ f ( x ) = 4∣ x ∣ − 7
f ∘ f ( x ) = ∣ x ∣
g ∘ g ( x ) = 16 x − 35

Answered by ElijahBenjaminCarter | 2025-07-06