Evaluate Mathematical 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) = √(x - 3)
Solution to Example 2
To find g(3), substitute x by 3 in the formula of the function
g (3) = √(3 - 3) = √(0) = 0
The domain of g is given by the interval
[3 , +infinity)
x = 0 is not included in the domain, hence
g(0) = √(0 - 3) = √(-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 3Evaluate 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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