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Other Functions and Reflections
Root Functions:
We will briefly consider root functions. These are functions of the form. The domain, range, and graph of these functions will depend on the index of the radical.
f ( ) x = nx
When n is 2, the function will be a square root function. The domain of a square root function is limited to only numbers greater than or equal to zero because it is not possible to take the square root of a negative number. The graph of f(x) = (^) x is shown below.
The graphs of other root functions with an even index will be similar to that of the square root function but only stretched flatter along the x-axis.
Root Functions (Continued):
When n is 3, the function will be a cube root function. The domain of a cube root function is not limited like the square root function and can be all real numbers. The graph of f(x) = is shown below.
(^3) x
Cubic Functions:
A cubic function is a power function with a degree power of 3. The domain of a cubic function is all real numbers because the cubic function is a polynomial function, which are continuous curves. The graph of f(x) = x^3 is shown below.
Reciprocal Functions (Continued):
For the function , the denominator cannot be zero. Also, since the numerator is a constant the
function will never be equal to zero. It will, however, approach zero as x gets larger in both the
negative and positive directions.
x
Reflections:
Functions can be reflected about the x-axis or y-axis by the multiplication of -1. Where the multiplication occurs will determine which axis the graph is reflected about. Let’s use the square root function as an example of reflection. If we multiply the radicand by -1, we would have the function − x.
f ( ) x = x
g x ( ) = − x
Since the -1 is being multiplied to x inside the radicand this would result in a reflection about the y-axis because the signs on the x-values will be change.
Reflections (Continued):
Reflection about the y-axis.
f ( ) x = x g x ( ) = − x
Now if the -1 is multiplied to the original function then we would end up with a reflection about the x-axis. Since the -1 is being multiplied to the original function the values that are being changed are the y values.
f ( ) x = x g x ( ) = − x
Reflection about the x-axis.
