Functions

Before solving function problems, it's essential to understand some key terms. Below are the definitions commonly used when discussing functions:

• Function: A rule that assigns each element of one set to a unique element of another set.
• Independent Variable (Input): In the function $y = f(x)$, $x$ is the independent variable.
• Dependent Variable (Output): In the function $y = f(x)$, $y$ is the dependent variable.
• Domain: The set of all possible input values for the function.
• Range: The set of all possible output values for the function.

Functions are often represented with lowercase letters like $f$, $g$, or $h$. For example, $f(x)$ denotes the function of $x$, meaning "f of x."

Think of a function as a process that takes an input $x$ and produces an output by substituting $x$ into the function rule. In simpler terms: $f(\text{input}) = \text{output}$. Importantly, each input corresponds to only one output, though multiple inputs can produce the same output. For example, if $h(x) = |x - 5|$, then $h(6) = 1$ and $h(4) = 1$.

To evaluate a function, substitute the given value for $x$ into the function. For instance, consider the function $f(x) = 2x^2 + 3$. To find $f(2)$, substitute $2$ for $x$:

$f(2) = 2(2)^2 + 3 = 2(4) + 3 = 8 + 3 = 11$

Thus, when $x = 2$, $f(x) = 11$. This substitution is written as $f(2) = 11$, meaning "the value of $f$ at $x = 2$ is 11."

Consider the following problem:

If $g(x) = 2x^2 + 17$, what is $g(12)$?

To solve, substitute $12$ for $x$:

$g(12) = 2(12)^2 + 17 = 2(144) + 17 = 288 + 17 = 305$

The correct answer is $305$.

The domain of a function is the set of all possible input values ($x$) for which the function is defined. The range is the set of all possible output values ($f(x)$). Below are some rules to determine the domain and range:

Domain Rules

The domain of a function includes all real numbers except those that:

• Result in division by zero (e.g., $\frac{1}{x-2}$ is undefined when $x = 2$).
• Result in an even root of a negative number (e.g., $\sqrt{-4}$ is not a real number).

For example, consider the function $f(x) = \frac{4}{x-2}$. Since the denominator cannot be zero, $x \neq 2$. Thus, the domain is all real numbers except $2$.

Range Rules

The range of a function is limited by the following:

• Absolute values cannot be negative.
• Even exponents cannot produce negative numbers.
• Square roots cannot return negative numbers.

For example, the function $g(x) = \sqrt{x - 2} + 1$ has a range of $g(x) \geq 1$, since the square root term is always non-negative.

Functions are a fundamental concept in mathematics, and understanding their terminology, domain, and range is crucial for solving related problems. By following the rules and practicing substitution, you can confidently tackle function-related questions.