1.2

Finding Domain and Range of a Function

• Algebraic Solution

Note: Because the expression is under a radical it must end up being ≥ 0 otherwise it would be imaginary.

Step 1. Set the equation as  ≥ 0

             ≥ 0

Step 2. Square both sides to get rid of the radical. It should end up being 

            x-4 ≥ 0

Step 3. Isolate the variable

          x-4+4≥ 0+4

                  x ≥ 4

x is all real numbers greater than or equal to zero [4,∞)

• Graphical Solution

f(x)=

The graph for this function looks like this

To find the domain and range simply look at the real line

As you an see the range is all real numbers greater than or equal to zero & the range is all real numbers greater than or equal to 4

Even and Odd Functions

Function 1

Find if f(x) = –3x² + 4 even, odd, or neither.

Plug in f(-x) and solve

f(–x) = –3(–x)² + 4 

      = -3(-x)(-x) +4

         = –3(x²) + 4          = –3x² + 4

The function is the same as it started out as therefore it's an even function, the graph would end up looking like this with the y axis as the line of symmetry

Function 2

Is f(x) = 2x³ – 4x even, odd, or neither?

Same as last time, plug in f(–x)

f(–x) = 2(–x)³ – 4(–x) 

         = 2(–x³) + 4x          = –2x³ + 4x

As you can see the result is the exact opposite of the original, which means that the function is odd. When graphed it would look like this with the point of origin as the line of symmetry 

Function 3

Is f(x) = 2x³ – 3x² – 4x + 4 odd, even or neither?

Plug in f(–x)

f(–x) = 2(–x)³ – 3(–x)² – 4(–x) + 4          = 2(–x³) – 3(x²)  + 4x + 4          = –2x³ –3x² + 4x + 4

It's not the same or opposite of the original equation which means it is neither and a graph for it would not follow the symmetry on the y axis or the point of origin.