Functions and Their Properties: College Algebra Review

Functions and Their Properties

Evaluating Functions at Indicated Values

Evaluating a function means substituting a given value for the variable and simplifying the result. This is a foundational skill in algebra, as it allows us to determine the output of a function for specific inputs.

• Key Point 1: Substitute the given value into the function wherever the variable appears.

• Key Point 2: Simplify the resulting expression to find the function's value at that input.

• Example: If \( f(x) = x^2 + 3 \), then \( f(2) = 2^2 + 3 = 7 \).

The Difference Quotient

The difference quotient is a formula that measures the average rate of change of a function over an interval. It is foundational for calculus but also important in algebra for understanding how functions change.

• Definition: The difference quotient for a function \( f(x) \) is given by:

$\frac{f(x+h) - f(x)}{h},\ h \neq 0$

• Key Point 1: Compute \( f(x+h) \) by replacing every \( x \) in the function with \( x+h \).

• Key Point 2: Subtract \( f(x) \) from \( f(x+h) \), then divide by \( h \).

• Example: For \( f(x) = 5 - 2x \),

$f(x+h) = 5 - 2(x+h) = 5 - 2x - 2h$ $\frac{f(x+h) - f(x)}{h} = \frac{[5 - 2x - 2h] - [5 - 2x]}{h} = \frac{-2h}{h} = -2$

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Determining the domain is essential for understanding where a function can be used.

• Key Point 1: For rational functions, exclude values that make the denominator zero.

• Key Point 2: For even roots (like square roots), exclude values that make the radicand negative.

• Example: For \( f(x) = \frac{x}{x-3} \), the domain is all real numbers except \( x = 3 \).

Determining Whether a Graph is a Function

A graph represents a function if and only if every vertical line intersects the graph at most once (the vertical line test).

• Key Point 1: If any vertical line crosses the graph more than once, it is not a function.

• Key Point 2: Equations can sometimes be solved for \( y \) in terms of \( x \) to check if they define functions.

• Example: The graph of \( y = x^2 \) is a function, but the graph of \( x = y^2 \) is not.

Analyzing Graphs of Functions

Analyzing a function's graph involves identifying intercepts, intervals of increase and decrease, and local maxima and minima.

• Key Point 1: Intercepts: Points where the graph crosses the axes.

• Key Point 2: Increasing/Decreasing Intervals: Where the graph rises or falls as you move left to right.

• Key Point 3: Local Maxima/Minima: Highest or lowest points in a local region of the graph.

• Example: For a parabola opening upwards, the vertex is a local minimum.

Even, Odd, and Neither Functions

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Function: \( f(-x) = f(x) \) for all \( x \) in the domain. Graph is symmetric about the y-axis.

• Odd Function: \( f(-x) = -f(x) \) for all \( x \) in the domain. Graph is symmetric about the origin.

• Neither: If neither condition holds, the function is neither even nor odd.

• Example: \( f(x) = x^2 \) is even; \( f(x) = x^3 \) is odd.

Graphing Using Transformations

Transformations shift or stretch the graph of a function. Understanding these helps in quickly sketching graphs of related functions.

• Key Point 1: Horizontal Translation: \( f(x - h) \) shifts the graph right by \( h \) units.

• Key Point 2: Vertical Translation: \( f(x) + k \) shifts the graph up by \( k \) units.

• Key Point 3: Reflection: \( -f(x) \) reflects the graph over the x-axis; \( f(-x) \) reflects over the y-axis.

• Example: \( y = (x-2)^2 + 1 \) is the graph of \( y = x^2 \) shifted right 2 units and up 1 unit.