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Chapter 4: Problem 1

Explain what it means when an equation is written in the form \(y=f(x)\)

Short Answer

Expert verified
When an equation is written in the form \(y = f(x)\), it indicates that y is a function of x, and the value of y depends on the input value x. A function is a mathematical relationship where each input (x) is associated with exactly one output (y), and f(x) represents the rule or transformation mapping x to y. Functions can be evaluated, represented graphically as a curve or line, illustrating the mathematical relationship between the variables x and y.

Step by step solution

01

Definition of Functions

A function is a mathematical relationship between two sets of numbers (or other mathematical objects) where each number in the first set (called the domain) is associated with exactly one number in the second set (called the range). In other words, a function is a rule that takes an input (x) and transforms it into an output (y).
02

Writing Equations in the Form y = f(x)

When an equation is written in the form y = f(x), it denotes that y is a function of x. This means that the value of y depends on the value of the input x. In this notation, f(x) represents the rule or transformation that maps x to the corresponding y value. For example, an equation can be given in the form \(y = 2x + 3\). In this case, f(x) is represented by \(2x + 3\).
03

Evaluating Functions

To evaluate a function, you simply replace x with the specific value and perform the operations described by the function. For example, given the function \(y = 2x + 3\), if you want to find the value of y when x is equal to 4, you would replace x with 4 and calculate the result: \(y = 2(4) + 3 = 11\).
04

Graphing Functions

A function can be represented graphically as a curve (or sometimes, a straight line) in a two-dimensional coordinate system. The horizontal axis typically represents the values of the independent variable (x), while the vertical axis represents the values of the dependent variable (y). Each point on the curve represents a unique pairing of x and y according to the function rule f(x). For example, if the function is given by \(y = 2x + 3\), the graph of the function will be a straight line with a slope of 2 and a y-intercept at 3. In summary, when an equation is written in the form y = f(x), it indicates that y is a function of x, meaning the value of y depends on the input value x.

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