Lesson 7: Chapter Summary and Review

New Concept

A function is a rule that assigns exactly one output to each input. We'll explore how to describe functions using equations like $f(x) = y$, tables, graphs, and words, and how to evaluate them for specific values.

What’s next

You've got the core concept. Next, you'll work through interactive examples and practice cards to master evaluating functions and identifying them in different forms.

Property

A function can be described in words, by a table, by a graph, or by an equation.

Function Notation.
Input variable → $f(x) = y$ ← Output variable

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.

Property

The Vertical Line Test.
A graph represents a function if and only if every vertical line intersects the graph at most one point.

Examples

• The set of points ${(2, 3), (4, 1), (5, 3)}$ represents a function because each $x$-value corresponds to only one $y$-value. No vertical line would cross more than one point.

• The set of points ${(1, 4), (1, -4), (3, 9)}$ does not represent a function. A vertical line at $x=1$ would pass through both $(1, 4)$ and $(1, -4)$, failing the test.

Property

Absolute Value.
The absolute value of $x$ is defined by

Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.

Examples

• To evaluate $|-9|$, since $-9 < 0$, we use the second case of the definition: $|-9| = -(-9) = 9$.

• To evaluate the expression $10 - |5 - 8|$, first compute the operation inside the absolute value bars: $5 - 8 = -3$. Then take the absolute value: $|-3| = 3$. Finally, subtract: $10 - 3 = 7$.

Property

Two variables are directly proportional if the ratios of their corresponding values are always equal.

Direct Variation.
$y$ varies directly with $x$ if

where $k$ is a positive constant called the constant of variation.

Direct Variation with a Power.
$y$ varies directly with a power of $x$ if

where $k$ and $n$ are positive constants.

Property

Inverse Variation.
$y$ varies inversely with $x$ if

where $k$ is a positive constant.

Inverse Variation with a Power.
$y$ varies inversely with $x^n$ if

where $k$ and $n$ are positive constants.
If the product $yx^n$ is constant and $n$ is positive, then $y$ varies inversely with $x^n$.

Examples

• The time $t$ required to travel a fixed distance varies inversely with speed $s$. If it takes 4 hours at 50 mph, the constant is $k = ts = 4(50) = 200$. The formula is $t = \frac{200}{s}$.