/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 107 The graph of a function of \(x\)... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 107

The graph of a function of \(x\) cannot have symmetry with respect to the \(x\) -axis.

Short Answer

Expert verified
The statement is correct, a function cannot be symmetric about the x-axis because it would mean that one x value has multiple y values, which is against the definition of a function.

Step by step solution

01

Understanding Concept of Function

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
02

Understanding Vertical Line Test

To check if a graph represents a function, you can use the vertical line test. According to this test, a vertical line drawn anywhere on the graph of the relation should intersect the graph in at most one point.
03

Understanding X-axis Symmetry and Functions

If a graph has symmetry about the x-axis, then a vertical line can intersect the graph in more than one point. Hence, the graph would not pass the vertical line test and consequently will not represent a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Line Test
The vertical line test is a simple method used to determine whether a graph represents a function. It involves drawing vertical lines over the graph. If any vertical line crosses the graph at more than one point, the graph does not represent a function. This is because, in a function, each input value should relate to only one output value.

The vertical line test is important because it gives a visual way to comprehend if a relation between variables (often x and y) satisfies the criteria for being a function.
  • Draw a vertical line over different parts of the graph.
  • If any line touches the graph more than once, it fails the vertical line test.
  • If no line touches the graph more than once, the graph represents a function.
By applying this test, we can easily identify whether or not the depicted graph adheres to the definition of a function.
Function Graph
A function graph is a visual representation of relationships between inputs and outputs. It's a plot of points in the coordinate system where each input or x-value is paired with an output or y-value. This graphical representation can help in identifying the behavior and properties of a function.

Through the plotted points, one can observe characteristics like intercepts, intervals of increase or decrease, and approximate solutions to equations.
  • Graphs provide insights into how the output changes with the input.
  • Analyzing a graph can reveal patterns such as linearity or curvature.
  • Graphs can quickly show important aspects like slope and intercept.
Function graphs are crucial tools in mathematics, engineering, and sciences, making abstract concepts more tangible and interpretable.
X-axis Symmetry
Having x-axis symmetry means that for every point \( (x, y) \, \), there exists a corresponding point \( (x, -y) \, \), thus mirroring across the x-axis.聽

For graphs of functions, x-axis symmetry is generally not allowed, as this would mean failing the vertical line test. More than one output for a given input is forbidden in functions by definition.

In short, if a graph shows x-axis symmetry, it cannot be considered a function, as no binding relationship between each x-value and one y-value can be maintained.
  • X-axis symmetry results in the same y-value being replaced by its negative counterpart.
  • Such symmetry leads to multiple intersection points with a vertical line.
Understanding x-axis symmetry is important when interpreting or confirming the nature of a graph.
Input-Output Relation
In mathematics, understanding the input-output relation of a function is pivotal. A function is essentially a rule that tells you how to take an input value (usually denoted by x) and calculate a corresponding output (often denoted by y).

This relation is depicted as pairs, written as (x, y), showing how each specific input has a particular output. This is fundamental in defining what constitutes a function. For an expression to be a function:
  • Each input must produce exactly one output.
  • Different inputs can have the same output, but not vice-versa.
By having a clear understanding of the input-output relation, one grasps more concretely why certain graphs represent functions and others do not.

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Most popular questions from this chapter

Distance In Exercises 79 and 80 , use the result of Exercise 77 to find the distance between the point and line. Point: \((2,3)\) Line: \(4 x+3 y=10\)

Sketching a Graph of a Function In Exercises \(31-38,\) sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(f(x)=4-x\)

Sketching the Graph of a Trigonometric Function In Exercises \(55-66,\) sketch the graph of the function. $$y=1+\cos \left(x-\frac{\pi}{2}\right)$$

True or False? In Exercises 85 and \(86,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The lines represented by $$a x+b y=c_{1} \quad\( and \)\quad b x-a y=c_{2}$$ are perpendicular. Assume \(a \neq 0\) and \(b \neq 0\)

Ferris wheel. The model for the height \(h\) of a Ferris wheel car is \(h=51+50 \sin 8 \pi t\) where \(t\) is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when \(t=0\) . Alter the model so that the height of the car is 1 foot when \(t=0\) .
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