For the functions f(x)=x+2 and g(x)=x^2-4, find the...

For the functions $f(x)=x+2$ and $g(x)=x^{2}-4,$ find the indicated function and state its domain. Explain the mistake that is made in each problem. $f \circ g$ Solution: $$ \begin{aligned} f \circ g &=f(x) g(x) \\ &=(x+2)\left(x^{2}-4\right) \\ &=x^{3}+2 x^{2}-4 x-8 \end{aligned} $$ Domain: $(-\infty, \infty)$ This is incorrect. What mistake was made?

For the functions $f(x)=x+2$ and $g(x)=x^{2}-4,$ find the indicated function and state its domain. Explain the mistake that is made in each problem.
$f \circ g$
Solution:
$$
\begin{aligned}
f \circ g &=f(x) g(x) \\
&=(x+2)\left(x^{2}-4\right) \\
&=x^{3}+2 x^{2}-4 x-8
\end{aligned}
$$
Domain: $(-\infty, \infty)$
This is incorrect. What mistake was made?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Domain of Composite Functions

The domain of a composite function, f(g(x)), consists of all the values of x that do not cause any issues in the intermediate steps. That is, x must be in the domain of g, and g(x) must lie in the domain of f. This concept ensures that the composite function is well-defined for all valid inputs, unlike misinterpreting operations which might lead to incorrect domain assumptions.

Function Composition

Function composition is the process of applying one function to the results of another. When we write f ? g, it means we first take the output of g(x) and then use that as the input for f(x); it is mathematically represented as f(g(x)). This is distinct from multiplying functions together and requires substitution rather than algebraic multiplication.

Distinction Between Composition and Multiplication of Functions

A common error when dealing with functions is to mistake composition (f ? g) for multiplication (f(x) * g(x)). The notation f ? g specifically instructs to substitute one function into the other, not to multiply their outputs. This distinction is critical for correctly evaluating expressions involving functions and obtaining the correct result.

Please give Ace some feedback