Composite Functions

Question 1: Notation of Function Composition

The Question: Let $f(x)$ and $g(x)$ be two functions. Which of the following options is(are) INCORRECT?

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

Detailed Solution:

This question is a direct test of the definition. The notation $(f \circ g)(x)$ means “apply f to the result of g(x)”.

• First Option: $(f \circ g)(x) = f(g(x))$. This is the correct definition.
• Second Option: $(g \circ f)(x) = g(f(x))$. This is also the correct definition for the reverse composition.
• Third Option: $(f \circ g)(x) = g(f(x))$. This is INCORRECT. It reverses the order of the functions.
• Fourth Option: $(g \circ f)(x) = f(g(x))$. This is INCORRECT. It also reverses the order.

Final Answer: The incorrect options are:

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

Question 2: Properties of Composite Functions

The Question: Suppose $f(x)$ and $g(x)$ are well defined functions. Which of the following statements is(are) CORRECT?

• For any given functions $f(x)$ and $g(x)$, $(f \circ g)(x) = (g \circ f)(x)$.
• The domain of a composite function $(f \circ g)(x)$ is always a subset of the domain of the function $g(x)$.
• The domain of a composite function $(f \circ g)(x)$ is always equal to the domain of the function $g(x)$.
• The range of a composite function $(f \circ g)(x)$ is always a subset of the range of the function $f(x)$.

Detailed Solution:

• First statement: This is FALSE. Function composition is not commutative. For example, if $f(x)=x+1$ and $g(x)=x^2$, then $f(g(x))=x^2+1$, but $g(f(x))=(x+1)^2$. These are not equal.

• Second statement: This is TRUE. To calculate $f(g(x))$, $x$ must first be a valid input for the inner function $g(x)$. The domain might be further restricted if the output $g(x)$ is not a valid input for $f(x)$, but it can never be larger than the domain of $g(x)$.

• Third statement: This is FALSE. The domain can be a proper subset (smaller than), but not always equal. For example, if $g(x)=x-5$ (domain is $\mathbb{R}$) and $f(x)=\sqrt{x}$ (domain is $x \ge 0$), the domain of $(f \circ g)(x)=\sqrt{x-5}$ is $x \ge 5$, which is a subset of, but not equal to, the domain of $g(x)$.

• Fourth statement: This is TRUE. The final outputs of the composite function $f(g(x))$ are produced by the outer function $f$. Therefore, the set of all possible outputs (the range of the composite) must be a subset of all possible outputs of $f$.

Final Answer: The correct options are:

• The domain of a composite function $(f \circ g)(x)$ is always a subset of the domain of the function $g(x)$.
• The range of a composite function $(f \circ g)(x)$ is always a subset of the range of the function $f(x)$.

Question 3: Computing Composite Functions

The Question: Suppose $f(x) = 8x$, $g(x) = 50 - 8x$, $h(x) = 50$, and $k(x) = 50 - 64x$ are functions. Which of the following options is true?

• $(f \circ g)(x) = k(x)$
• $(g \circ f)(x) = k(x)$
• $(g \circ f)(x) = h(x)$
• $(f \circ g)(x) = h(x)$

Detailed Solution:

We need to compute both $(f \circ g)(x)$ and $(g \circ f)(x)$ and compare them to $h(x)$ and $k(x)$.

1. Compute $(f \circ g)(x) = f(g(x))$

• Start with the outer function: $f(x) = 8x$.
• Replace the $x$ in $f(x)$ with the entire expression for $g(x)$:
  • $f(g(x)) = 8(g(x))$
  • $f(g(x)) = 8(50 - 8x)$
  • $f(g(x)) = 400 - 64x$
• This result does not match $h(x)$ or $k(x)$.

2. Compute $(g \circ f)(x) = g(f(x))$

• Start with the outer function: $g(x) = 50 - 8x$.
• Replace the $x$ in $g(x)$ with the entire expression for $f(x)$:
  • $g(f(x)) = 50 - 8(f(x))$
  • $g(f(x)) = 50 - 8(8x)$
  • $g(f(x)) = 50 - 64x$
• This result is exactly the definition of $k(x)$.

Final Answer: The correct option is $(g \circ f)(x) = k(x)$.