Math Help

Description of Functions of One Independent Variable

Example: Find the range of the function y = x³ + 3 x + 5

if the domain is ˗ 4 ≤ x < 2 .

The domain are all the numbers between ˗ 4 and +2 including ˗ 4 but not including +2.

y(˗ 4) = ˗ 47 y(+2) = 19 The range of y(x) is ˗ 47 ≤ y(x) < 19 . (result)

Example: Find the domain of the function

f(x) = √ (x² + 4 ) / [ √ (25 ˗ 4x) ˗ √ ( x + 9 ) ]

from √ (x² + 4 ), x² + 4 ≥ 0 for all values of x so no restriction on the domain

from √ (25 ˗ 4x) 25 ˗ 4x ≥ 0 so ˗ 4x ≥ ˗ 25, Thus x ≤ 25/4

from √ ( x + 9 ) x + 9 ≥ 0 x ≥ ˗ 9

Now the denominator is zero when [ √ (25 ˗ 4x) ˗ √ ( x + 9 ) ] = 0

or when √ (25 ˗ 4x) = √ ( x + 9 )

(25 ˗ 4x) = ( x + 9 )

or 16 = 5 x and x = 16 / 5

Thus x ≠ 16 / 5

Finally the domain is: [ ˗ 9, 16/5 ) U (16/5, 25/4 ] (result)

Example: Determine if the following function, f(x), is even or odd.

f(x) = cos (x³) ˗ sin(x⁴)

Recall that if f(x) is odd then f( ˗ x) = ˗ f(x) and even if f(˗ x ) = f(x).

Here f(˗ x) = cos ( (˗x³) ) ˗ sin ( (˗x⁴) ) = cos (x³) ˗ sin ( (x⁴) = f(x)

Therefore f(x) is an even function. (result)