Does This Graph Show a Function Explain How You Know

Does This Graph Show a Function Explain How You Know

Learning Outcomes

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

Equally nosotros take seen in examples above, we can correspond a part using a graph. Graphs display many input-output pairs in a small infinite. The visual data they provide often makes relationships easier to understand. We typically construct graphs with the input values forth the horizontal axis and the output values along the vertical centrality.

The most mutual 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]x[/latex], or [latex]y=f\left(ten\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\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right)[/latex]. If the role is defined for only a few input values, then the graph of the function is but 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 us that [latex]f\left(0\correct)=2[/latex] and [latex]f\left(6\right)=1[/latex]. However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\correct)[/latex] and [latex]\left(6,i\right)[/latex] because the curve passes through those points.

vertical line examination
tin exist used to determine whether a graph represents a function. A vertical line includes all points with a detail [latex]x[/latex] value. The [latex]y[/latex] value of a point where a vertical line intersects a graph represents an output for that input [latex]ten[/latex] value. If we can draw
vertical line that intersects a graph more than once, then the graph does
define a function considering that [latex]x[/latex] value has more than than one output. A function has only one output value for each input value.

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Three graphs visually showing what is and is not a function.

How To: Given a graph, use the vertical line test to determine if the graph represents a function.

  1. Inspect the graph to run into if whatever vertical line drawn would intersect the bend more than once.
  2. If in that location is any such line, the graph does non represent a part.
  3. If no vertical line can intersect the curve more than once, the graph does stand for a office.

Case: Applying the Vertical Line Test

Which of the graphs correspond(s) a function [latex]y=f\left(x\correct)?[/latex]

Graph of a polynomial.

Try It

Does the graph below represent a function?

Graph of absolute value function.

The Horizontal Line Test

Once nosotros have adamant that a graph defines a function, an easy manner to determine if it is a one-to-one role is to employ the
horizontal line examination. Draw 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 we can describe
horizontal line that intersects a graph more than in one case, then the graph does
stand for a function because that [latex]y[/latex] value has more one input.

How To: Given a graph of a role, use the horizontal line examination to decide if the graph represents a 1-to-one function.

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

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 issues with them. When learning to read, we get-go with the alphabet. When learning to practise arithmetic, we start with numbers. When working with functions, it is similarly helpful to accept a base of operations set of building-block elements. We call these our “toolkit functions,” which form a prepare of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions nosotros will use [latex]10[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.

Nosotros will see 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 apace by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.

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

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Does This Graph Show a Function Explain How You Know