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Evaluating functions

There are certain common values of functions that you should memorize before you do the problems in this sectin. These values are often used in examples, and you need to be able to recognize them on sight. Section 14, topic 2 lists the values you should memorize.

Mental calculation

If an expression is not too complicated, you can somtimes substitute a number into a function mentally.

Examples
  1. Consider the function f(x) = (2/3)x^(-1/3)
    to find think to get
    f(1) "+1 raised to any power is still +1" (2/3)(1)^(-1/3) = 2/3
    f(0) "x to a negative power puts the x in the denominator, so this gives zero in the denominator" (2/3)(1/(0)^(1/3)) is undefined

  2. Consider the function f(x) = x^2 \sin(x)
    to find think to get
    f(pi) "sine is zero at pi, so the product will be zero" (pi)^2 sin(pi) = pi^2 (0) = 0

Formal substitution

When the calculations are more complicated, you may need to write out some steps.

Examples
For
f(x) = 3x^2 + 2x
find
(f(b) - f(a))/(b-a)
when a=-2 and b=1. Here is is important to use enough parentheses to keep the minus signs straight.
(f(1)-f(-2))/(1-(-2))=((3(1)^2+2(1))-(3(-2)^2+2(-2)))/3=(5-(3(4)-4))/3=(5-8)/3=-1
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
    (f(b) - f(a))/(b-a)
    when a=5 and b=6.
    (f(b) - f(a))/(b-a)   =  

    (Enter answer, then click: . Answer message: )
  2. Let
    f(x)   =   h x   +   x ( x   -   j )

    find
    (f(b) - f(a))/(b-a)
    when a=6 and b=1.
    (f(b) - f(a))/(b-a)   =  

    (Enter answer, then click: . Answer message: )
For more practice, click: .
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precal: 10.2 - evaluating functions
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