# Does This Graph Represent a Function Why or Why Not

### Learning Outcomes

• Verify a part using the vertical line test
• Verify a one-to-one part with the horizontal line test
• Identify the graphs of the toolkit functions

As we have seen in examples above, we can represent a function using a graph. Graphs brandish many input-output pairs in a small space. The visual data they provide often makes relationships easier to empathize. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical centrality.

The most common graphs name the input value $x$ and the output value $y$, and we say $y$ is a function of $x$, or $y=f\left(x\right)$ when the office is named $f$. The graph of the function is the set up of all points $\left(x,y\correct)$ in the plane that satisfies the equation $y=f\left(x\correct)$. If the function is defined for just a few input values, so the graph of the function is merely a few points, where the
10-coordinate of each point is an input value and the
y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in the graph beneath tell u.s.a. that $f\left(0\right)=2$ and $f\left(6\right)=1$. However, the fix of all points $\left(x,y\right)$ satisfying $y=f\left(x\right)$ is a curve. The bend shown includes $\left(0,2\right)$ and $\left(6,i\right)$ because the curve passes through those points.

The
vertical line test
tin be used to decide whether a graph represents a function. A vertical line includes all points with a item $x$ value. The $y$ value of a bespeak where a vertical line intersects a graph represents an output for that input $x$ value. If we can depict
whatsoever
vertical line that intersects a graph more than once, then the graph does
non
define a function because that $x$ value has more than one output. A office has only i output value for each input value.

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### How To: Given a graph, utilise the vertical line exam to determine if the graph represents a part.

1. Inspect the graph to run into if whatsoever vertical line fatigued would intersect the bend more than than once.
2. If there is any such line, the graph does non stand for a function.
3. If no vertical line can intersect the bend more once, the graph does represent a role.

### Example: Applying the Vertical Line Test

Which of the graphs correspond(s) a function $y=f\left(10\right)?$

### Try It

Does the graph beneath stand for a role?

## The Horizontal Line Examination

Once we have adamant that a graph defines a function, an easy way to make up one’s mind if it is a one-to-one function is to use the
horizontal line test. Depict horizontal lines through the graph. A horizontal line includes all points with a item $y$ value. The $x$ value of a point where a vertical line intersects a office represents the input for that output $y$ value. If we tin can describe
whatsoever
horizontal line that intersects a graph more once, and so the graph does
not
correspond a part because that $y$ value has more than one input.

### How To: Given a graph of a function, utilise the horizontal line exam to determine if the graph represents a 1-to-one function.

1. Audit the graph to see if any horizontal line drawn would intersect the bend more than once.
2. If there is whatsoever such line, the function is non 1-to-one.
3. If no horizontal line tin intersect the curve more than than one time, the office is ane-to-one.

## Identifying Basic Toolkit Functions

In this text we explore functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve bug with them. When learning to read, we commencement with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to take a base of operations set of building-cake elements. We call these our “toolkit functions,” which grade a ready of basic named functions for which nosotros know the graph, formula, and special properties. Some of these functions are programmed to private buttons on many calculators. For these definitions we will apply $x$ as the input variable and $y=f\left(ten\right)$ as the output variable.

Nosotros volition run across these toolkit functions, combinations of toolkit functions, their graphs, and their transformations often throughout this volume. It will exist very helpful if nosotros can recognize these toolkit functions and their features rapidly by proper noun, formula, graph, and basic tabular array properties. The graphs and sample table values are included with each part shown below.

Toolkit Functions
Proper name Function Graph
Abiding $f\left(ten\right)=c$, where $c$ is a constant
Identity $f\left(x\correct)=x$
Absolute value $f\left(x\right)=|x|$
Quadratic $f\left(x\right)={x}^{2}$
Cubic $f\left(ten\right)={10}^{3}$
Reciprocal/ Rational $f\left(x\right)=\frac{one}{ten}$
Reciprocal / Rational squared $f\left(10\right)=\frac{1}{{x}^{ii}}$
Square root $f\left(ten\correct)=\sqrt{10}$
Cube root $f\left(x\right)=\sqrt[3]{10}$

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