For example:
Step-by-step explanation:
Throughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, y = 4x +1 is an equation that represents a function. When you input values for x, you can determine a single output for y. In this case, if you substitute x = 10 into the equation you will find that y must be 41; there is no other value of y that would make the equation true.
Rather than using the variable y, the equations of functions can be written using function notation. Function notation is very useful when you are working with more than one function at a time, and substituting more than one variable in for x.
Some people think of functions as “mathematical machines.” Imagine you have a machine that changes a number according to a specific rule, such as “multiply by 3 and add 2” or “divide by 5, add 25, and multiply by −1.” If you put a number into the machine, a new number will pop out the other end, having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.
You can also call the machine “f” for function. If you put x into the box, f(x),comes out. Mathematically speaking, x is the input, or the “independent variable,” and f(x) is the output, or the “dependent variable,” since it depends on the value of x.
f(x)= 4x + 1 is written in function notation and is read “f of x equals 4x plus 1.” It represents the following situation: A function named f acts upon an input, x, and produces f(x) which is equal to 4x + 1. This is the same as the equation as y = 4x + 1.
Function notation gives you more flexibility because you don’t have to use y for every equation. Instead, you could use f(x) or g(x) or c(x). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.
You could write the formula for perimeter, P = 4s, as the function p(x) = 4x, and the formula for area, A = x2, as a(x) = x2. This would make it easy to graph both functions on the same graph without confusion about the variables.
Which two equations represent the same function?
A) y = 2x – 7 and f(x) = 7 – 2x
B) 3x = y – 2 and f(x) = 3x – 2
C) f(x) = 3x2 + 5 and y = 3x2 + 5
D) None of the above
Equations written using function notation can also be evaluated. With function notation, you might see a problem like this.
Given f(x) = 4x + 1, find f(2).
You read this problem like this: “given f of x equals 4x plus one, find f of 2.” While the notation and wording is different, the process of evaluating a function is the same as evaluating an equation: in both cases, you substitute 2 for x, multiply it by 4 and add 1, simplifying to get 9. In both a function and an equation, an input of 2 results in an output of 9.
f(x) = 4x + 1
f(2) = 4(2) + 1 = 8 + 1 = 9
You can simply apply what you already know about evaluating expressions to evaluate a function. It’s important to note that the parentheses that are part of function notation do not mean multiply. The notation f(x) does not mean f multiplied by x. Instead the notation means “f of x” or “the function of x” To evaluate the function, take the value given for x , and substitute that value in for x in the expression.
