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Chapter 1: Problem 1

Give four ways that functions may be defined and represented.

Short Answer

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Question: List and briefly describe four different ways functions can be defined and represented in mathematics. Answer: The four different ways functions can be defined and represented are: 1. Algebraic Expression: Functions can be represented using equations that show the relationship between input and output variables, such as f(x)=3x^2+4. 2. Graphs: Functions can be visually represented as curves or points on a Cartesian plane, illustrating the behavior of the function and key aspects like intercepts and maxima. 3. Tables: Functions can be represented using tables that pair input values with their corresponding output values, making it easier to identify patterns and relationships between them. 4. Verbal Description: Functions can be defined and described through written explanations, which is particularly helpful for real-world problems where context might be better understood with words instead of symbolic equations.

Step by step solution

01

Way 1: Algebraic Expression (Formula)

Algebraic expressions or formulas are commonly used to represent functions. In this method, a function is denoted by 'f(x)' and/or 'y' and is defined by an equation that represents the relation between the input variable, usually x, and the output variable, usually y. For example, the function f(x)=3x^2+4 represents a quadratic function where for each input x, the output is calculated using the given equation.
02

Way 2: Graphs

Graphs offer a visual representation of a function. The function is depicted as a curve or set of points on a Cartesian plane, where the x-axis represents the input values while the y-axis represents the output values. Graphs help to identify key aspects like intercepts, maxima, minima, and the general behavior of the function. For example, the graph of a linear function y = mx + b can be represented as a straight line with slope m and y-intercept b.
03

Way 3: Tables

Functions can also be represented using tables, where input values are paired with their corresponding output values in a well-organized manner. This representation makes it easier to see various number patterns and behavior for certain inputs and outputs. For example, a table of a linear function y = 2x + 1 might look like: x | y -1 | -1 0 | 1 1 | 3 2 | 5
04

Way 4: Verbal Description

A function can also be defined and represented through a verbal description. In this case, the relationship between the input and output variables is given through a written explanation. This method is particularly helpful when describing real-world problems where the context might be better understood with words instead of symbolic equations. For example, a verbal description of a function might be: "The cost of producing x items is 5 dollars per item plus a fixed cost of 100 dollars."

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Most popular questions from this chapter

Consider the general quadratic function \(f(x)=a x^{2}+b x+c,\) with \(a \neq 0\) a. Find the coordinates of the vertex in terms of \(a, b,\) and \(c\) b. Find the conditions on \(a, b,\) and \(c\) that guarantee that the graph of \(f\) crosses the \(x\) -axis twice.

Design a sine function with the given properties. It has a period of 12 hr with a minimum value of -4 at \(t=0 \mathrm{hr}\) and a maximum value of 4 at \(t=6 \mathrm{hr}\)

Given the following information about one trigonometric function, evaluate the other five functions. $$\csc \theta=\frac{13}{12} \text { and } 0<\theta<\pi / 2$$

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Verify that the function $$ D(t)=2.8 \sin \left(\frac{2 \pi}{365}(t-81)\right)+12 $$ has the following properties, where \(t\) is measured in days and \(D\) is measured in hours. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and \(9.2,\) respectively, which occur approximately at \(t=172\) and \(t=355\) respectively (corresponding to the solstices). c. \(\overline{D(81)}=12\) and \(D(264)=12\) (corresponding to the equinoxes).
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