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Function Notation: Intro / Evaluation (page 1 of 3)

Sections: Introduction & Evaluating at a number, Evaluating at a variable, Even and odd functions

You've been playing with "y =" sorts of equations for some time now. And you've seen that the "nice" equations (straight lines, say, rather than ellipses) are the ones that you can solve for "y =" and then plug into your graphing calculator. These "y =" equations are functions. But the question you are facing at the moment is "Why do I need this function notation, and how does it work?"

Think back to when you were in elementary school: Your teacher gave you worksheets containing statements like "[  ] + 2 = 4" and told you to fill in the box. Now that you're grown, your teacher will give you worksheets containing statements like "x + 2 = 4" and will tell you to "solve for x". Why did your teachers switch from boxes to variables? Well, think about it: How many shapes would you have to use for formulas like the one for the area A of a trapezoid with upper base a, lower base b, and height h?




    A = ( h/2 )(a + b)

If you try to express the above, or something more complicated, using variously-shaped boxes, you'd quickly run out of shapes. Besides, you know from experience that "A" stands for "area", "h" stands for "height", and "a" and "b" stand for the lengths of the parallel top and bottom sides. Heaven only knows what a square box or a triangular box might stand for. So they switched from boxes to variables because, while the boxes and the letters mean the exact same thing (namely, a slot waiting to be filled with a value), variables are better. Variables are more flexible, easier to read, and can give you more information.

The same is true of "y" and "f(x)" (pronounced as "eff-of-eks"). For functions, the two notations mean the exact same thing, but "f(x)" gives you more flexibility and more information. You used to say "y = 2x + 3; solve for y when x = –1". Now you say "f(x) = 2x + 3; find f(–1)" (pronounced as "f-of-x is 2x plus three; find f-of-negative-one"). You do exactly the same thing in either case: you plug in –1 for x, multiply by 2, and then add the 3, simplifying to get a final value of +1.

But function notation gives you greater flexibility than using just "y" for every formula. Your graphing calculator will list different functions as y1, y2, etc. In textbooks and when writing things out, we use names like f(x), g(x), h(x), s(t), etc. With this notation, you can now use more than one function at a time without confusing yourself or mixing up the formulas, wondering "Okay, which 'y' is this, anyway?" And the notation can be usefully explanatory: "A(r) = (pi)r2" indicates the area of a circle, while "C(r) = 2(pi)r" indicates the circumference. Both functions have the same plug-in variable (the "r"), but "A" reminds you that this is the formula for "area" and "C" reminds you that this is the formula for "circumference".   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

How do we go about evaluating functions? First, remember this: While parentheses have, up until now, always indicated multiplication, the parentheses do not indicate multiplication in function notation. The expression "f(x)" means "plug a value for x into a formula f "; the expression does not mean "multiply f and x"! Don't embarrass yourself by pronouncing (or thinking of) "f(x)" as being "f times x".

In function notation, the "x" in "f(x)" is called "the argument of the function", or just "the argument". So if they give you "f(2)" and ask for the "argument", the answer is just "2".

Let's proceed to "evaluation": You evaluate "f(x)" just as you would evaluate "y".

  • Given  f(x) = x2 + 2x – 1, find  f(2).

      f(2) = (2)2 +2(2) – 1
             = 4 + 4 – 1
             = 7

  • Given  f(x) = x2 + 2x – 1, find  f(–3).

      f(–3) = (–3)2 +2(–3) – 1
               = 9 – 6 – 1
               = 2

(If you experience difficulties with negatives, try using parentheses as I just did above; it helps keep track of things like whether or not the exponent is on the "minus" sign.) Remember that "x" is just a box, waiting for something to be put into it. Don't let odd-looking problems scare you:

    Well, f(@) = (@)2 + 2(@) – 1 = @2 + 2@ – 1.

The above example is kinda pointless, I'll grant you, but it clearly illustrates how the notation works.

Now YOU try!

An important type of function is called a "piecewise" function, so called because, well, it's in pieces:

    f(x) = 2x^2 – 1 for x < 1, x + 4 for x >= 1

As you can see, this function is split into two halves: the half that comes before x = 1, and the half that goes from x = 1 to infinity. Which half of the function you use depends on what the value of x is. If we want to evaluate f(0), we must, since 0 < 1, use the first half of the function. Then f(0) = 2(0)2 – 1 = 0 – 1 = –1. If we want to evaluate f(2), then we must, since 2 > 1, use the second half of the function. Then f(2) = (2) + 4 = 6. If we want to evaluate f(1), we must also use the second half, because the first half is only for x's that are strictly less than 1. The second half is for x's that are greater than or equal to1. Then f(1) = (1) + 4 = 5.

Now YOU try!

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Cite this article as:

Stapel, Elizabeth. "Function Notation: Introduction / Evaluating at a Number." Purplemath. Available from Accessed



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