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. $$\begin{aligned}(f+g)(2) &=\left(x+2+x^{2}-4\right)(2) \\&=\left(x^{2}+x-2\right)(2) \\&=2 x^{2}+2 x-4\end{aligned}$$ 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.
$$\begin{aligned}(f+g)(2) &=\left(x+2+x^{2}-4\right)(2) \\&=\left(x^{2}+x-2\right)(2) \\&=2 x^{2}+2 x-4\end{aligned}$$
Domain: $(-\infty, \infty)$
This is incorrect. What mistake was made?

Key Concepts

Substitution Mistake

A common mistake in function evaluation is to mistakenly treat the function input like a factor that multiplies the entire expression, rather than correctly substituting the value into each term. The error often occurs when the substitution is performed after manipulating the algebraic form in a way that changes the end result; the correct approach is to substitute the value into the function immediately, thus avoiding any improper scaling of the terms.

Determining the Domain of Function Operations

The domain of a function is the set of all input values for which the function is defined. When creating a new function through operations such as addition, the domain of the resulting function is typically the intersection of the domains of the individual functions. It is crucial to carefully check that the operation does not inadvertently introduce any restrictions or extraneous factors that might limit the domain.

Function Evaluation

This concept refers to the process of substituting a specific input value into a function and then carrying out the operations within the function as defined by its rule. Correct evaluation means that you replace the variable with the given number and compute the result, without misapplying any additional multiplications or operations that are not indicated by the function’s definition.

Addition of Functions

When two functions are added together, the resulting function is defined by the sum of the corresponding outputs for any given input. It is important to form the proper combination of the functions’ expressions and then evaluate the resulting expression correctly at any specified input, rather than mixing up the order of operations or misinterpreting the function notation.

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