Functions and Function Notation

Section 3.1: Functions and Function Notation

Learning Objectives

• Determine whether a relation represents a function.

• Find the value of a function.

• Use the vertical line test to identify functions.

• Graph the functions listed in the library of functions.

Ordered Pairs and Relations

• Ordered Pair: Shown as (first component, second component). Examples include $(x,y)$, $(5, -2)$, $(m,3)$, $(even,8)$ .

• Relation: A set (collection) of ordered pairs. The symbol ${}$ is used to show a set.

• Domain: The set of all first components of the ordered pairs.

  • Each value in the domain is called an input or independent variable.

• Range: The set of all second components of the ordered pairs.

  • Each value in the range is called an output or dependent variable.

Example 1: Analyzing a Relation

Given the relation: ${(1,5), (-5,2), (0,0), (1,-3)}$

i. Finding the Domain:
* Collect the first components: $\lbrace 1, -5, 0 \rbrace$ .
* Note: Even though $1$ is the input of two ordered pairs ($(1,5)$ and $(1,-3)$), it is only written once in the domain.
* The values in the domain, ordered from smallest to largest, are: $\lbrace -5, 0, 1 \rbrace$.
* These values ( $-5, 0, 1$ ) are the inputs or independent variables of the relation.

ii. Finding the Range:
* Collect the second components: $\lbrace 5, 2, 0, -3 \rbrace$.
* If any value was repeated, it would only be written once.
* The values in the range, ordered from smallest to largest, are: $\lbrace -3, 0, 2, 5 \rbrace$.
* These values ( $-3, 0, 2, 5$ ) are the outputs or dependent variables of the relation.

Think-Pair-Share 1:

Given the relation: ${(-7, -2), (3,0), (8, -2), (0,12), (-15,9)}$

• How many ordered pairs?: $\text{5 ordered pairs}$.

• Domain: Make sure to use correct symbols.

  • First components: $\lbrace -7, 3, 8, 0, -15 \rbrace$.

  • Ordered domain: $\lbrace -15, -7, 0, 3, 8 \rbrace$.

  • The values in the domain are called inputs or independent variables.

• Range: Make sure to use correct symbols.

  • Second components: $\lbrace -2, 0, 12, 9 \rbrace$.

  • Ordered range: $\lbrace -2, 0, 9, 12 \rbrace$.

  • The values in the range are called outputs or dependent variables.

What is a Function?

• Function: A relation in which each input value leads to exactly one output value.

  • This means that none of the input values is repeated.

• The domain of a function is the set of all its inputs.

• The range of a function is the set of all its outputs.

Function Notation

• The notation $y = f(x)$ defines a function named $f$.

  • $x$ is the independent variable (or input).

  • $y$ is the dependent variable (or output).

  • It is read as "$y$ is a function of $x$".

• Example: $y = f(x)$

  • $f$ is the name of the function.

  • $x$ is the input or the independent variable.

  • $y$ or $f(x)$ is the output or the dependent variable.

• Example: $y = h(m)$

  • Reads as "$y$ is a function of $m$".

  • $h$ is the name of the function.

  • $m$ is the input or the independent variable.

  • $y$ or $h(m)$ is the output or dependent variable.

• Example: $s = v(t)$

  • Reads as "$s$ is a function of $t$".

  • $v$ is the name of the function.

  • $t$ is the input or independent variable.

  • $s$ or $v(t)$ is the output or dependent variable.

Example 2: Identifying Functions

• The relation ${(p,m), (q,n), (r,n)}$ is a function because none of the inputs ($p, q, r$) is repeated.

• The relation ${(p,x), (q,y), (r,z)}$ is a function because none of the inputs ($p, q, r$) is repeated.

• The relation ${(p,x), (q,y), (q,z)}$ is NOT a function because the input $q$ is repeated (it leads to two different outputs, $y$ and $z$).

Think-Pair-Share 2:

$k = g(c)$ reads as k is a function of c; so, g is the name of the function, $c$ is the input or independent variable, and k is the output or dependent variable.

Example 3: Function in a Real-World Context (Coffee Shop Menu)

a) Is "price" a function of "item"?
* Here, "item" is the input (independent value) and "price" is the output (dependent value).
* Since none of the inputs (Plain Donut, Jelly Donut, Chocolate Donut) is repeated, price IS a function of item.

b) Is "item" a function of "price"?
* Here, "price" is the input (independent value) and "item" is the output (dependent value).
* Since the input value, $1.99$, repeats (it corresponds to both Plain Donut and Chocolate Donut), item is NOT a function of price.

Representing a Function

• A function can be represented using a table of values.

  • Example Table:
    | Input (x) | Output (y) |
    | :-------- | :--------- |
    | 1 | 5 |
    | -5 | 2 |
    | 0 | 0 |
    | 4 | -3 |

• A function can be represented in formula form.

  • Examples: $f(x) = 3x^2 - 7$; $g(t) = 3t - t^2 + 5$

• A function can be represented using a graph.

  • A graph represents a function if it passes the vertical line test.

    • Vertical Line Test: There is no vertical line that intersects the graph at more than one point.

Think-Pair-Share 3: Which of the following represents a function?

• a) (Graph): This graph represents a function. Any vertical line would intersect the graph at most once.

• b) $y = 4 - x^2$: This is a function. For every $x$ input, there is only one $y$ output.

• c) $x^2 + y^2 = 6$: This is NOT a function. For example, if $x=0$, then $y^2=6$, so $y = \pm \sqrt{6}$. A single input ($x=0$) leads to two outputs, violating the definition of a function. Graphically this is a circle, which fails the vertical line test.

• d) (Graph of a parabola opening sideways): This graph is NOT a function. A vertical line could intersect the graph at two points (e.g., for positive $x$ values, there would be a positive and negative $y$ value).

Finding Input and Output Values of a Function

i. From a Function Formula

• If the input value is given and output value is needed:

  1. Substitute the input variable in the formula with the given value.

  2. Calculate the output.

• If the output value is given and input value is needed:

  1. Substitute the output variable in the formula with the given value.

  2. Calculate the input.

Example 4: Evaluating a Function Formula

Given $f(x) = x^3 - 6x + 1$, evaluate $f(3)$.

• Substitute the input value $3$ into the formula:
  $f(3) = (3)^3 - 6(3) + 1$
  $f(3) = 27 - 18 + 1$
  $f(3) = 10$

• This means with the input $3$, the output value is $10$. The ordered pair $(3,10)$ is a point on the graph of $f(x)$.

Think-Pair-Share 4: Using the function from Example 4 ($f(x) = x^3 - 6x + 1$)

a) $f(-2) = (-2)^3 - 6(-2) + 1 = -8 + 12 + 1 = 5$
* That means, with the input $-2$, output value is $5$. The ordered pair $(-2,5)$ is a point on the graph of $f(x)$.

b) $f(a) = a^3 - 6a + 1$

Example 5: Solving for Input Given Output

Given $g(p) = \sqrt{3 + 2p}$, solve for $g(p) = 3$.

1. Substitute $g(p)$ with $3$:
   $3 = \sqrt{3 + 2p}$

2. Square both sides of the equation to eliminate the square root:
   $(3)^2 = (\sqrt{3 + 2p})^2$
   $9 = 3 + 2p$

3. Solve for $p$:
   $9 - 3 = 2p$
   $6 = 2p$
   $\frac{6}{2} = \frac{2p}{2}$
   $3 = p$

Think-Pair-Share 5: Solving for Input Given Output

Given $h(m) = m^2 - 5$, solve for $h(m) = 4$.

1. Substitute $h(m)$ with $4$:
   $4 = m^2 - 5$

2. Solve for $m$:
   $4 + 5 = m^2$
   $9 = m^2$
   $m = \pm \sqrt{9}$
   $m = \pm 3$

ii. From a Function Given in Tabular Form

• If input value is given:

  1. Find it in the row or column of input values.

  2. Identify the corresponding output value that is paired with the given input value.

• If output value is given:

  1. Find it in the row or column of output values.

  2. Identify the corresponding input value(s) that are paired with the given output value. (There could be more than one corresponding input value).

Example 6: Using a Table

Given the function $f(x)$ in tabular form:

| x | -3 | -1 | 0 | 4 | 5 |
| :--- | :- | :- | :- | :- | :- |
| f(x) | 5 | 1 | 1 | 5 | 8 |

a) Find $f(0)$:
* In the column of inputs, find $0$. The corresponding output value is $1$.
* Therefore, $f(0) = 1$.

b) Solve for $f(x) = 5$:
* In the column of outputs, find $5$. The corresponding input values are $-3$ and $4$.
* Therefore, $f(x) = 5$ has two solutions: $x = -3, x = 4$.

Think-Pair-Share 6: Using a Table

Given function $g(n)$ below:

| n | -3 | -1 | 0 | 3 | 5 |
| :--- | :- | :- | :- | :- | :- |
| g(n) | 8 | 2 | 7 | 1 | 8 |

a) $g(3)$: Find $3$ in the input row. The corresponding output is $1$. Therefore, $g(3) = 1$.

b) $g(1)$: Find $1$ in the input row. There is no input $1$ present in the table. Therefore, $g(1)$ is undefined from this table.

c) Solve for $g(n) = 8$: Find $8$ in the output row. The corresponding input values are $-3$ and $5$.
* Therefore, $g(n) = 8$ has two solutions: $n = -3, n = 5$.

d) Solve for $g(n) = 7$: Find $7$ in the output row. The corresponding input value is $0$.
* Therefore, $g(n) = 7$ has one solution: $n = 0$.

iii. From a Graph

• If input value is given:

  1. Find it on the horizontal (x-axis) axis.

  2. Find the corresponding point on the graph.

  3. Find the point's coordinate on the vertical (y-axis) axis (this is the output).

• If output value is given:

  1. Find it on the vertical (y-axis) axis.

  2. Find the corresponding point(s) on the graph.

  3. Find the point's coordinate(s) on the horizontal (x-axis) axis (this is the input).

Example 7: Using a Graph

Use the graph of $f(x)$ (assuming a graph is provided with points like $( -1, 4), (0, 1), (2, 1), (3, 2)$ etc.)

a) $f(-1)$:
* Locate $-1$ on the x-axis. Move up to the graph and then horizontally to the y-axis.
* (Based on typical examples, if a point exists at $x=-1$ and $y=4$)
* Therefore, $f(-1) = 4$.

b) Solve $f(x) = 1$:
* Locate $1$ on the y-axis. Move horizontally to intersect the graph, then vertically down to the x-axis.
* (Based on typical examples, if points exist at $x=0$ and $x=2$ for $y=1$)
* Therefore, there are two input values that correspond to the output value $1$: $x = 0$ and $x = 2$.

Think-Pair-Share 7: Using a Graph

Use the graph of $f(x)$ (assuming a graph is provided, e.g., similar to Example 7 with extended points)

a) $f(-2)$: (Assuming a point on the graph, e.g., $(-2,0)$). Therefore, $f(-2) = 0$.

b) $f(4)$: (Assuming a point on the graph, e.g., $(4,2)$). Therefore, $f(4) = 2$.

c) $f(x) = 1$: (Assuming points at $x=0$ and $x=2$ for $y=1$). Therefore, $x=0, x=2$.

d) $f(x) = 2$: (Assuming points at $x=-3$ and $x=3$ for $y=2$). Therefore, $x=-3, x=3$.

Summary

• How to determine if a relation is a function?

  • From a set of ordered pairs: Check if any input (first component) is repeated. If an input is repeated with different outputs, it is not a function.

  • From a table: Check if any input value in the input column/row has more than one corresponding output value. If so, it is not a function.

  • From a graph: Apply the Vertical Line Test. If any vertical line intersects the graph at more than one point, it is not a function.

• In $g = h(p)$, what is the independent variable/input? What is the dependent variable/output?

  • Independent variable/Input: $p$

  • Dependent variable/Output: $g$ or $h(p)$

• If $f(x) = 3x^2 - 7$, how do we find $f(-2)$? And how do we solve $f(x) = 5$?

  • To find $f(-2)$: Substitute $-2$ for $x$ in the formula: $f(-2) = 3(-2)^2 - 7 = 3(4) - 7 = 12 - 7 = 5$.

  • To solve $f(x) = 5$: Set the function formula equal to $5$ and solve for $x$: $3x^2 - 7 = 5 \implies 3x^2 = 12 \implies x^2 = 4 \implies x = \pm 2$.

Toolkit Functions

• Table 13 in section 3.1 of the textbook contains graphs of toolkit functions that must be learned. (Specific details are not in the transcript, but this objective is listed.)