Which Graph Represents the Function Y 2x 4

As we take seen in examples higher up, we tin represent a office using a graph. Graphs display many input-output pairs in a small infinite. The visual information they provide often makes relationships easier to understand. We typically construct graphs with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a part of [latex]10[/latex], or [latex]y=f\left(10\right)[/latex] when the function is named [latex]f[/latex]. The graph of the function is the set of all points [latex]\left(x,y\correct)[/latex] in the aeroplane that satisfies the equation [latex]y=f\left(x\right)[/latex]. If the function is defined for merely a few input values, so the graph of the function is just a few points, where the
x-coordinate of each signal is an input value and the
y-coordinate of each point is the corresponding output value. For case, the black dots on the graph in the graph below tell usa that [latex]f\left(0\right)=2[/latex] and [latex]f\left(vi\right)=1[/latex]. However, the set of all points [latex]\left(10,y\right)[/latex] satisfying [latex]y=f\left(10\right)[/latex] is a curve. The curve shown includes [latex]\left(0,ii\right)[/latex] and [latex]\left(6,i\right)[/latex] because the bend passes through those points.

The
vertical line exam
can be used to determine whether a graph represents a function. A vertical line includes all points with a particular [latex]10[/latex] value. The [latex]y[/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]x[/latex] value. If nosotros can describe
whatsoever
vertical line that intersects a graph more than than in one case, and then the graph does
not
define a function because that [latex]ten[/latex] value has more one output. A function has only one output value for each input value.

The Horizontal Line Test

Once we take determined that a graph defines a function, an easy style to determine if it is a one-to-i function is to utilize the
horizontal line exam. Describe horizontal lines through the graph. A horizontal line includes all points with a particular [latex]y[/latex] value. The [latex]ten[/latex] value of a point where a vertical line intersects a function represents the input for that output [latex]y[/latex] value. If nosotros can draw
whatsoever
horizontal line that intersects a graph more one time, then the graph does
not
represent a function because that [latex]y[/latex] value has more than 1 input.

Identifying Basic Toolkit Functions

In this text nosotros 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 offset with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base of operations set of edifice-block elements. Nosotros call these our “toolkit functions,” which form a set of bones named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to private buttons on many calculators. For these definitions nosotros will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.

We volition see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if nosotros tin recognize these toolkit functions and their features chop-chop by proper noun, formula, graph, and basic tabular array properties. The graphs and sample table values are included with each function shown below.

Toolkit Functions
Name	Office	Graph
Constant	[latex]f\left(10\right)=c[/latex], where [latex]c[/latex] is a constant
Identity	[latex]f\left(x\right)=ten[/latex]
Absolute value	[latex]f\left(x\correct)=|x|[/latex]
Quadratic	[latex]f\left(x\right)={x}^{2}[/latex]
Cubic	[latex]f\left(x\right)={x}^{3}[/latex]
Reciprocal/ Rational	[latex]f\left(x\right)=\frac{1}{10}[/latex]
Reciprocal / Rational squared	[latex]f\left(x\right)=\frac{one}{{x}^{2}}[/latex]
Foursquare root	[latex]f\left(x\right)=\sqrt{x}[/latex]
Cube root	[latex]f\left(10\right)=\sqrt[3]{ten}[/latex]

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