# Which Function is Represented by the Graph

Which Function is Represented by the Graph

### Learning Outcomes

• Verify a role using the vertical line examination
• Verify a 1-to-one function with the horizontal line test
• Place the graphs of the toolkit functions

As we accept seen in examples above, nosotros can represent a function using a graph. Graphs display many input-output pairs in a small space. The visual information they provide frequently makes relationships easier to understand. Nosotros typically construct graphs with the input values along the horizontal axis and the output values forth the vertical axis.

The about common graphs name the input value $x$ and the output value $y$, and we say $y$ is a office of $ten$, or $y=f\left(x\right)$ when the function is named $f$. The graph of the function is the set of all points $\left(10,y\right)$ in the plane that satisfies the equation $y=f\left(x\right)$. If the function is defined for only a few input values, then the graph of the office is only 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 usa that $f\left(0\right)=two$ and $f\left(half-dozen\right)=1$. However, the set of all points $\left(ten,y\correct)$ satisfying $y=f\left(10\right)$ is a curve. The curve shown includes $\left(0,2\right)$ and $\left(6,1\right)$ considering the curve passes through those points.

The
vertical line test
can be used to determine whether a graph represents a office. A vertical line includes all points with a particular $x$ value. The $y$ value of a point where a vertical line intersects a graph represents an output for that input $x$ value. If we tin can draw
any
vertical line that intersects a graph more than once, so the graph does
not
ascertain a function considering that $x$ value has more than than ane output. A function has only one output value for each input value.

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

1. Inspect the graph to run across if any vertical line drawn would intersect the bend more than once.
2. If at that place is any such line, the graph does not stand for a role.
3. If no vertical line can intersect the curve more than once, the graph does represent a function.

### Example: Applying the Vertical Line Examination

Which of the graphs represent(due south) a function $y=f\left(10\right)?$

### Attempt It

Does the graph below stand for a part?

## The Horizontal Line Test

Once nosotros have determined that a graph defines a function, an easy way to decide if information technology is a one-to-one function is to use the
horizontal line test. Draw horizontal lines through the graph. A horizontal line includes all points with a particular $y$ value. The $10$ value of a point where a vertical line intersects a function represents the input for that output $y$ value. If we tin can depict
whatsoever
horizontal line that intersects a graph more than once, and then the graph does
not
represent a role because that $y$ value has more than one input.

### How To: Given a graph of a office, use the horizontal line test to determine if the graph represents a 1-to-one role.

1. Audit the graph to see if any horizontal line drawn would intersect the bend more than in one case.
2. If there is any such line, the role is not one-to-ane.
3. If no horizontal line can intersect the curve more than than once, the office is one-to-ane.

## 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 problems with them. When learning to read, nosotros start with the alphabet. When learning to practise arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base ready of building-block elements. We telephone call these our “toolkit functions,” which form a ready of basic named functions for which we know the graph, formula, and special backdrop. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will employ $10$ as the input variable and $y=f\left(x\right)$ as the output variable.

Nosotros will run across these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this volume. It will be very helpful if we tin can recognize these toolkit functions and their features quickly past proper noun, formula, graph, and basic table backdrop. The graphs and sample table values are included with each role shown beneath.

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

## Contribute!

Did you take an idea for improving this content? Nosotros’d love your input.