Evaluate Functions

Evaluate real valued functions: A step by step tutorial, with examples and detailed solutions.




To find the value f(a) of a function, a has to be in the domain of f. In what follows, we are considering only real valued functions.

Example 1: Evaluate, if possible, f(-2) and f(2) given that f is defined by

f (x) = - 4 / ( x + 2)

Solution to Example 1

  • Function f given above has domain

    (-infinity , -2) U (-2 , +infinity)

  • Since at x = -2 the denominator of f(x) is equal to 0,

    f(-2) = undefined.

  • To find f(2), substitute x by 2 in f(x) = -4 / ( x + 2)

    f(2) = - 4 / (2 + 2) = -1.


Example 2: Evaluate, if possible, g(3) and g(0) given that g is defined by

g (x) = sqrt(x - 3) , sqrt means square root.

Solution to Example 2

  • To find g(3), substitute x by 3 in the formula of the function

    g (3) = sqrt(3 - 3) = sqrt(0) = 0

  • The domain of g is given by the interval

    [3 , +infinity)

  • x = 0 is not included in the domain, hence

    g(0) = sqrt(0 - 3) = sqrt(-3) = not a real number.

Example 3: Evaluate, if possible, h(4), g(4) and h(4) / g(4) where functions h and g are defined by

h (x) = 3x - 8 , g (x) = x 2 - 16

Solution to Example 3

  • Evaluate h(4)

    h(4) = 3(4) - 8 = 4

  • Evaluate g(4)

    g (4) = 4 2 - 16
    = 16 -16 = 0

  • In evaluating h(4) / g(4), g(4) which is the denominator is equal to 0. In mathematics division by zero is not allowed. Hence

    h(4) / g(4) = undefined


Example 4: Evaluate, if possible, h(t -1) where function h is defined by

h (x) = 2 x 2 - 2 x + 2

Solution to Example 4

  • The domain of this function is the set of all real numbers. Hence h(t -1) is given by

    h (t - 1) = 2 (t - 1) 2 - 2 (t - 1) + 2

  • Expand the square and group like terms

    h (t - 1) = 2 (t 2 - 2t + 1) - 2t + 2 + 2

    = 2t 2 - 4t + 2 - 2t + 4

    = 2t 2 - 6t + 6


Exercises:

1 - Evaluate f(9) given that f(x) = 2 x 2 + 2

2 - Evaluate g(1), h(1) and g(1) / h(1) given that g(x) = x 3 + 1 and h(x) = x - 1

3 - Evaluate f(t + 2) given that f(x) = - 2 x 2 + 2x



Solutions to Above Exercises:

1 - f(9) = 164

2 - g(1) = 2 , h(1) = 0 , g(1) / h(1) = undefined

3 - f(t + 2) = - 2 t 2 - 6t - 4
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Updated: 2 April 2013

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