6. Algebra Review

Intro to Functions and Notation

6. Algebra Review

Intro to Functions and Notation: Videos & Practice Problems

Video Lessons

Concept

Intro to Relations and Functions

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Intro to Relations and Functions Video Summary

A relation is a connection between inputs (x-values) and outputs (y-values), often represented as a set of ordered pairs in the form $(x, y)$. To understand relations, you can list all the input values and output values separately and then show how each input corresponds to an output. For example, given ordered pairs like $(-2, 2)$, $(1, 1)$, $(3, -2)$, and $(1, 4)$, the inputs are $-2$, $1$, and $3$, while the outputs are $2$, $1$, $-2$, and $4$. Notice that the input $1$ corresponds to two different outputs, $1$ and $4$.

A function is a special type of relation where each input is paired with at most one output. This means no input value maps to more than one output value. For instance, if you have ordered pairs like $(-4, 2)$, $(-2, -1)$, $(1, 2)$, and $(3, 4)$, each input corresponds to exactly one output. Here, $-4$ maps to $2$, $-2$ maps to $-1$, $1$ maps to $2$, and $3$ maps to $4$. Since no input repeats with different outputs, this relation qualifies as a function.

It is important to recognize that all functions are relations, but not all relations are functions. This distinction hinges on whether each input has a unique output.

Relations and functions can be represented in various ways beyond ordered pairs. They can be shown as correspondences, tables, graphs, or equations. For example, a function can be expressed as an equation like $y = 2x + 3$, which defines a unique output $y$ for each input $x$.

Understanding the difference between relations and functions is fundamental in mathematics, as it helps in analyzing how variables depend on each other and in identifying valid functions for modeling real-world scenarios.

Example

Intro to Relations and Functions Example 1

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Intro to Relations and Functions Example 1 Video Summary

To determine whether a relation is a function, it is essential to analyze the mapping between inputs (often represented as x-values) and outputs (y-values). A function is defined as a relation where each input corresponds to exactly one output. This means no single input can map to multiple outputs.

For example, consider a relation given in a table format. Begin by listing the unique input values. Then, observe the outputs associated with each input. If any input value is paired with more than one output value, the relation is not a function. For instance, if the input ½ maps to both 4 and 5, this violates the definition of a function because one input has multiple outputs.

Alternatively, when given a mapping diagram with inputs and outputs represented by letters or symbols, the exact numerical values of outputs are not necessary to determine if the relation is a function. The key is to verify that each input connects to only one output. If every input has a single corresponding output, the relation qualifies as a function.

In summary, the fundamental criterion for a function is that each input value must have at most one output value. This concept is crucial in understanding functions in algebra and helps distinguish functions from general relations.

Concept

Function Notation

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Function Notation Video Summary

Functions can be expressed using a special notation called function notation, which provides a clear way to represent the relationship between inputs and outputs. Instead of writing an equation as y = 3x - 1, we use the notation f(x) = 3x - 1, where f is the name of the function and x is the input variable. This notation is read as "f of x" and represents the output value corresponding to the input x.

To evaluate a function at a specific input, such as f(4), you substitute the input value into the function's expression. For example, given f(x) = 3x - 1, substituting x = 4 yields:

\[f(4) = 3 \times 4 - 1 = 12 - 1 = 11\]

This means when the input is 4, the output of the function is 11. Function notation is versatile and can use different letters to name functions, such as g(x) or h(x), but the process of substitution and evaluation remains the same.

Functions can also be represented as sets of ordered pairs, where each pair consists of an input (x-value) and its corresponding output (y-value). For example, if a function g is given by the set of ordered pairs, to find g(1), you locate the pair where the input is 1 and identify the output. If the pair is (1, 5), then g(1) = 5.

When functions are graphed, the input corresponds to the x-coordinate, and the output corresponds to the y-coordinate on the graph. To find h(-2) for a function h(x) graphed as a curve, locate the point where x = -2 on the x-axis, then find the corresponding y-value on the curve. If the y-value is 3, then h(-2) = 3.

Understanding function notation is essential for working with functions in algebra and calculus. It allows you to clearly express and evaluate functions, whether given by equations, ordered pairs, or graphs. Remember, the key idea is that the function takes an input x and produces an output, often denoted as f(x), g(x), or h(x), depending on the function's name.

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Problem

Find $f of negative 3$ for the following functions.
(A) $f (x) = 2 x^{2} - 4 x + 1$

$31$

$29$

$34$

$25$

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Problem

Find $f of negative 3$ for the following functions.
(B) $f (x) = \frac{x + 4}{x - 1}$

$\frac{1}{2}$

$\frac{1}{4}$

$- \frac{1}{4}$

$- \frac{1}{2}$

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Example

Function Notation Example 2

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2m

Function Notation Example 2 Video Summary

To define a function f(x) that expresses y as a function of x, the key step is to solve the given equation for y. This process involves isolating y on one side of the equation, which then allows you to rewrite the equation in the form y = f(x). For example, consider the equation 3x - 2y = 8. First, subtract 3x from both sides to get -2y = 8 - 3x. Next, divide every term by -2 to isolate y, resulting in y = $\frac{8 - 3x}{-2}$. Simplifying by distributing the negative sign gives y = $\frac{3x - 8}{2}$. This expression defines the function f(x) = $\frac{3x - 8}{2}$.

Similarly, for the equation 4x + 5y = 15, isolate y by subtracting 4x from both sides, yielding 5y = 15 - 4x. Dividing both sides by 5 gives y = $\frac{15 - 4x}{5}$. This defines the function f(x) = $\frac{15 - 4x}{5}$.

By solving for y and expressing it explicitly in terms of x, you create a function f(x) that maps each input x to a corresponding output y. This method ensures the equation represents a function, which is fundamental in understanding relationships between variables in algebra. The process highlights the importance of algebraic manipulation skills such as adding or subtracting terms on both sides and dividing by coefficients to isolate variables.

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Example

Function Notation Example 3

Video duration:
4m

Function Notation Example 3 Video Summary

The given data represents the percentage of the US population using the Internet over various years, which can be analyzed as a function. In this context, the years serve as the input values (domain), and the corresponding Internet usage percentages are the output values (range). Since each year corresponds to exactly one percentage value, this relationship satisfies the definition of a function, where every input has a unique output.

To identify the domain, list all the input years: 2000, 2005, 2010, 2015, 2020, and 2022. These represent the specific points in time for which Internet usage data is available. The range consists of the output values, which are the percentages of Internet users for each year: 43.1, 68, 74, 88.5, 90, and 92. These values indicate the proportion of the population using the Internet at each corresponding year.

When defining this function as f, where f(x) represents the percentage of Internet users in year x, evaluating f(2015) means finding the output when the input is 2015. According to the data, f(2015) = 88.5, indicating that 88.5% of the US population used the Internet in 2015.

Conversely, if given an output value such as f(x) = 92, the task is to determine the input year x that corresponds to this percentage. From the data, 92% Internet usage occurred in 2022, so x = 2022. This demonstrates how function notation can be used to interpret and analyze real-world data by linking inputs and outputs clearly.

Understanding how to determine whether a table defines a function, identifying the domain and range, and evaluating or solving for inputs and outputs using function notation are fundamental skills in interpreting data relationships. These concepts are essential for analyzing trends and making predictions based on tabular data.

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Concept

Vertical Line Test

Video duration:
3m

Vertical Line Test Video Summary

A function is a special type of relation where each input corresponds to exactly one unique output. To quickly determine whether a relation represented on a graph is a function, the vertical line test is an effective method. This test involves imagining or drawing vertical lines across the graph. If any vertical line intersects the graph at more than one point, the relation is not a function. Conversely, if every vertical line touches the graph at most once, the relation qualifies as a function.

When applying the vertical line test, consider a graph with discrete points. By drawing vertical lines through each point, if no line crosses more than one point, the graph represents a function. For example, a set of plotted points where each vertical line intersects only one point confirms the relation is a function.

In contrast, for graphs depicting smooth curves with infinitely many points, the vertical line test remains applicable. If any vertical line intersects the curve at multiple points, this indicates that a single input corresponds to multiple outputs, violating the definition of a function. For instance, a vertical line crossing a curve at two points shows the relation is not a function.

Understanding the vertical line test is essential for analyzing functions graphically. It provides a quick and reliable way to verify whether a relation meets the criteria of a function by ensuring each input has a unique output. This concept is fundamental in mathematics, especially when studying functions and their properties.

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Problem

State the inputs and outputs of the following relation. Is it a function?
${(- 3, 5), (0, 2), (3, 5)}$
$Two labeled ovals side by side: green for Inputs and yellow for Outputs, used to analyze a relation's function status.$

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Problem

State the inputs and outputs of the following relation. Is it a function?
$the set open paren 2 comma 5 close paren comma open paren 0 comma 2 close paren comma open paren 2 comma 9 close paren$
$Two labeled ovals side by side: green oval labeled "Inputs" and yellow oval labeled "Outputs".$

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Example

Vertical Line Test Example 4

Video duration:
1m

Vertical Line Test Example 4 Video Summary

To determine whether a graph represents a function, the vertical line test is a useful method. This test states that if a vertical line can be drawn anywhere on the graph and it intersects the graph at more than one point, then the graph is not a function. Conversely, if every vertical line drawn intersects the graph at most once, then the graph is indeed a function.

For example, consider three different graphs:

1. **Graph A**: When applying the vertical line test, any vertical line drawn across this graph intersects it at only one point. This consistent behavior confirms that Graph A is a function.

2. **Graph B**: In this case, a vertical line can be drawn that intersects the graph at multiple points. This indicates that Graph B does not satisfy the criteria for a function.

3. **Graph C**: Similar to Graph A, any vertical line drawn on Graph C intersects it at only one point. Thus, Graph C is also classified as a function.

In summary, Graphs A and C are functions, while Graph B is not. This understanding of the vertical line test is essential for identifying functions in various mathematical contexts.

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Intro to Functions and Notation: Videos & Practice Problems

Intro to Relations and Functions

Intro to Relations and Functions Video Summary

Intro to Relations and Functions Example 1

Intro to Relations and Functions Example 1 Video Summary

Function Notation

Function Notation Video Summary

Find f(−3)f(-3) for the following functions.(A) f(x)=2x2−4x+1f\(\left\)(x\(\right\))=2x^2-4x+1

Find f(−3)f(-3) for the following functions.(B) f(x)=x+4x−1f\(\left\)(x\(\right\))=\(\frac{x+4}{x-1}\)

Function Notation Example 2

Function Notation Example 2 Video Summary

Function Notation Example 3

Function Notation Example 3 Video Summary

Vertical Line Test

Vertical Line Test Video Summary

State the inputs and outputs of the following relation. Is it a function?{(−3,5),(0,2),(3,5)}\(\left\]\lbrace\[\left\)(-3,5\(\right\)),\(\left\)(0,2\(\right\)),\(\left\)(3,5\(\right\))\(\right\]\rbrace\)

State the inputs and outputs of the following relation. Is it a function?{(2,5),(0,2),(2,9)}\(\left\]\lbrace\[\left\)(2,5\(\right\)),\(\left\)(0,2\(\right\)),\(\left\)(2,9\(\right\))\(\right\]\rbrace\)

Vertical Line Test Example 4

Vertical Line Test Example 4 Video Summary

Your Liberal Arts Math tutors

Find $f of negative 3$ for the following functions.
(A) $f (x) = 2 x^{2} - 4 x + 1$

Find $f of negative 3$ for the following functions.
(B) $f (x) = \frac{x + 4}{x - 1}$

State the inputs and outputs of the following relation. Is it a function?
${(- 3, 5), (0, 2), (3, 5)}$
$Two labeled ovals side by side: green for Inputs and yellow for Outputs, used to analyze a relation's function status.$

State the inputs and outputs of the following relation. Is it a function?
$the set open paren 2 comma 5 close paren comma open paren 0 comma 2 close paren comma open paren 2 comma 9 close paren$
$Two labeled ovals side by side: green oval labeled "Inputs" and yellow oval labeled "Outputs".$