Functions, Domains, and Ranges

Functions, Domains, and Ranges

Definition of a Function

• A function is a correspondence between input numbers (usually x values) and output numbers (usually y values).
• Each input number is sent to exactly one output number.
• A function can be thought of as a rule or a machine where x values are input and y values are output.

Example

• The "biological mother" function: Input is any person, and the output is their biological mother.
• This is a function because each person has only one biological mother.
• The "mother" function (sending each person to their mother) is NOT a function because some people may have multiple mothers (biological, adoptive, stepmother).

Function Notation

• Functions are often described with equations.
• Example: $y = x^2 + 1$
• Alternative notation: $f(x) = x^2 + 1$
• $f(x)$ represents the output value (y).
• It is important to note that $f(x)$ does not represent multiplication of $f$ by $x$.

Evaluating Functions

• To evaluate $f(x)$, plug in a value for $x$.
• Example:
  • $f(2) = 2^2 + 1 = 5$
  • $f(5) = 5^2 + 1 = 26$
• Evaluating with more complex expressions:
  • $f(a + 3) = (a + 3)^2 + 1$
  • $f(a + 3) = a^2 + 6a + 9 + 1 = a^2 + 6a + 10$
• Importance of parentheses when plugging in for $x$: Without parentheses, you would not be squaring the whole expression.

Functions Described by Graphs

• A function can be described by a graph instead of an equation.
• Not all graphs represent functions.

Vertical Line Test

• The graph of a circle is not a function because it violates the vertical line test.
• Vertical Line Test: A vertical line intersects the graph in more than one point.
• If any vertical line intersects the graph at most one point, the graph represents a function.
• This is because every $x$ value will have at most one $y$ value that corresponds to it.

Evaluating Functions from Graphs

• To evaluate $g(2)$, find $x = 2$ on the x-axis and find the corresponding $y$ value on the graph.
• Example: If at $x = 2$, the y value is 3, then $g(2) = 3$.
• If there is no point on the graph for a given $x$ value, then the function is undefined for that $x$ value.
• Example: If there is no point on the graph at $x = 5$, then $g(5)$ is undefined or does not exist.

Domain and Range

• Domain: All possible $x$ values that make sense for the function.
• Range: All possible $y$ values that make sense for the function.
• If an $x$ value does not have a corresponding $y$ value, it is not in the domain.

Finding Domain and Range from a Graph

• Domain: Take the "shadow" or projection of the graph onto the $x$-axis to see what $x$ values are hit.
  • Express the domain using interval notation.
  • Example: If the graph hits all $x$ values from -8 to 4, including the endpoints, the domain is $[-8, 4]$.
• Range: Take the "shadow" or projection of the graph onto the $y$-axis to see what $y$ values are hit.
  • Express the range using interval notation.
  • Example: If the graph hits all $y$ values from -5 to 3, including the endpoints, the range is $[-5, 3]$.

Finding Domain Algebraically

• One way to find the domain and range is to graph the function.
• It's often possible to find the domain more quickly by using algebraic considerations.
• Consider what $x$ values can be plugged into the expression without resulting in an impossible calculation.

Restrictions on Domain

1. Exclude $x$ values that make the denominator zero (since division by zero is undefined).
2. Exclude $x$ values that make an expression inside a square root sign negative (since the square root of a negative number is not real).
   • This applies to any even root. Odd roots (e.g., cube root) can accept negative numbers.
3. Later, logarithmic functions will introduce additional exclusions.

Examples

Example A

• Function: Presence of a denominator. Need to exclude x values that make the denominator zero.
  • Solve $x^2 - 4x + 3 = 0$
  • Factor: $(x - 3)(x - 1) = 0$
  • Solutions: $x = 3$ and $x = 1$
  • These values must be excluded from the domain.
  • Domain in interval notation: $(-\infty, 1) \cup (1, 3) \cup (3, \infty)$

Example B

• Function: Presence of a square root. Need to exclude $x$ values that make the expression inside the square root negative.
  • Condition: $3 - 2x \geq 0$
  • Solve: $3 \geq 2x \implies x \leq \frac{3}{2}$
  • Domain in interval notation: $(-\infty, \frac{3}{2}]$

Example C

• Function: Involves both a square root and a denominator. Two conditions must be considered:
  1. The denominator cannot be zero.
  2. The expression inside the square root must be greater than or equal to zero.
• From the previous example, we know:
  • $x \neq 3$ and $x \neq 1$
  • $x \leq \frac{3}{2}$
• Combine these conditions on a number line.
• The domain includes numbers that satisfy BOTH conditions.
• Domain in interval notation: $(-\infty, 1) \cup (1, \frac{3}{2}]$

Summary: Discussed functions, function evaluation, domain, and range.