Rational and Radical Functions

Radical Functions

Radical and Power Functions

Radical functions have an inverse relationship with power functions.
A radical function is one whose rule can be written using an expression with a variable under a root, such as a square root or cube root:
f(x)=p(x)nf(x)=\sqrt[\scriptsize{n}]{p(x)}
There is an inverse relationship with power functions that have whole-number powers for simple radical functions of the form:
f(x)=xnf(x)=\sqrt[\scriptsize{n}]{x}
It is a reflection of the graph across the line y=xy=x of:
f(x)=xnf(x)=x^n
For even functions, the inverse relationship holds only for values of the power function with x0x\geq0.

Power Functions and Radicals

Even Power Functions and Radicals Odd Power Functions and Radicals

A function and its inverse are reflections of each other across the line y=xy=x. Whether the inverse of a power function of the form f(x)=xnf(x)=x^n is a function depends on the value of nn. If nn is positive and even, the inverse is not a function unless the domain is restricted to x0x\geq0. If nn is positive and odd, the inverse is a function, and the domain does not need to be restricted.

Graphing Radical Functions

Radical functions are graphed using a variety of techniques, including transformations of the radical parent functions and technology.

Radical functions can be difficult to graph by hand. Even when radical functions are graphed using technology, it can be useful to analyze the functions to understand the properties of the graphs.

For some calculators, it may be difficult to enter roots other than square and cube roots. In such cases, it may be necessary to use rational exponents. Taking the nnth root of an expression is the same as raising it to the power of 1n\frac{1}{n}:
xn=x1n\sqrt[\scriptsize{n}]{x}={x^{\footnotesize{\frac{1}{n}}}}
For example:
f(x)=3x2+4x14f(x)=\sqrt[\scriptsize{4}]{3x^2+4x-1}
To graph the function, it may need to be entered as:
(3x2+4x1)14(3x^2+4x-1)^{\footnotesize{\frac{1}{4}}}
Step-By-Step Example
Graph a Radical Function
Graph and then analyze the function:
f(x)=x24x5f(x)=\sqrt{x^2-4x-5}
Step 1

Look at the expression under the radical. Values of xx that make this expression negative are not in the domain of ff because the square root of a negative value is not a real number.

Graph the given function together with:
g(x)=x24x5g(x)=x^2-4x-5
gg is quadratic and can be factored as:
(x5)(x+1)(x-5)(x+1)
The zeros are –1 and 5. The values are negative for 1<x<5-1<x<5. So, the domain is restricted to values outside that interval.
Step 2

Identify the domain and range of ff.

The domain is x1x\leq-1 and x5x\geq5.

The graph shows that all the yy-values are on or above the xx-axis, so the range is y0y\geq0.

Solution

Analyze the function. Identify maxima (highest points), minima (lowest points), and zeros. Describe where the function is increasing and decreasing.

The graph shows the minimum value of zero, which occurs at x=1x=-1 and x=5x=5. There is no maximum.

The zeros of the function are also at x=1x=-1 and x=5x=5.

The function is decreasing for x<1x<-1 and increasing for x>5x>5.