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Function Notation

Using function notation

It is easier to say "f(x) is linear," or "f(x) = mx + b" than to say "you get y by multiplying x by some constant and then adding some other constant to your answer." The language of function notation gives a name, f for the operation, and at the same time identifies the independent variable x.

Examples
  1. If f(x)=x^2+6, then f(0)=0^2+6, f(-3)=(-3)^2+6, f(t)=t^2+6, f(a+h)=(a+h)^2+6, and f(_)=(_)^2+6, for any _
  2. If f(x)=sqrt(x), then f(3.7)=sqrt(3.7), f(2+h)=sqrt(2+h), f(2)+h=sqrt(2)+h, f(2a)=sqrt(2a), and 2f(a)=2sqrt(a)
    (Note that f(2+h) is not the same as f(2)+h, and f(2a) is not the same as 2f(a).)

Interpreting the meaning of expressions using function notation

When we use function notation, it is crucial to "translate" the meaning of the notation correctly. Similar-appearing forms can have very different meanings.

Examples
  1. "Translate" the following function sentences:
    Let f(x)=(x-1)^3   This tells the rule for forming the function f using the variable x.
    Solve f(x)=27   This tells you to find the x value which makes y=27. In other words, solve the equation
    (x-1)^3 = 27
    (to get x = 4).
    Find f(3)   This tells you to find the y value corresponding to x=3, or
    (3-1)^3 = 8

  2. Let
    f(x) = x^2+3x-10
    Find the y-intercept of the graph of f and the places where the graph crosses the x-axis.
    (1) To find the y-intercept, we are looking for where the graph crosses the y axis, which occurs when x=0. Thus we're looking for the value
    f(0) = (0)^2 + 3(0) - 10 = -10
    This gives us a y-intercept of (0,-10).
    (2) The graph will cross the x-axis whenever y=0, which is where f(x)=0. Thus we want
    x^2+3x-10=0, or (x+5)(x-2)=0, so x=-5 or x=2
    The graph crosses the x-axis at (-5,0) and (2,0).
  3. Let
    f(x) = 4e^x
    Find x so that f(x)=12.
    We want
    4e^x=12, or e^x=3, so ln(e^x)=ln(3), and x=ln(3).
Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Let
    f(x)   =  
    a x + b

    c x + 1

    Find y coordinate of the y-intercept: y=
    If x = -3, what is f(x)?
    if f(x) = 1, find x:
    (Enter answers, then click: . Answer message: )
  2. Let
    f(x)   =   h x   +   x ( x   -   j )

    Find the x value(s) of the x-intercept(s) (leave one blank if there are more spaces than you need) and
    What are the y value(s) of the intercept(s)? and
    If x = 3, find f(x):
    (Enter answer, then click: . Answer message: )
For more practice, click: .
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precal: 10.1 - function notation
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©2003-2007 Gavin LaRose, Pat Shure / University of Michigan Math Dept. / Regents of the University of Michigan