2. Chapter 4: Rational, Power, and Root Functions
4.1 Rational Functions and Graphs
4.2 More on Rational Functions and Graphs
4.3 Rational Equations, Inequalities, Models, and Applications
4.4 Functions Defined by Powers and Roots
4.5 Equations, Inequalities, and Applications Involving Root Functions

3. 4.4 Functions Defined by Powers and Roots
f (x) = xp/q, p/q in lowest terms
if q is odd, the domain is all real numbers
if q is even, the domain is all nonnegative real numbers
Power and Root Functions
A function f given by f (x) = xb, where b is a constant, is a power function. If , for some integer n  2, then f is a root function given by f (x) = x1/n, or equivalently, f (x) =

4. 4.4 Graphing Power Functions
Example Graph f (x) = xb, b = 0.3, 1, and 1.7, for
x  0.
Solution The larger values of b cause the graph of
f to increase faster.

5. 4.4 Applying Properties of Rational Exponents
Example Simplify each expression by hand.
(a) (b)
Solution
(a)
(b)

6. 4.4 Modeling Wing Size of a Bird
Example Heavier birds have larger wings with more surface
area. For some species of birds, this relationship can be
modeled by S (x) = 0.2x2/3, where x is the weight of the bird in
kilograms and S is the surface area of the wings in square
meters. Approximate S(0.5) and interpret the result.
Solution
The wings of a bird that weighs 0.5 kilogram have a surface
area of about square meter.

7. 4.4 Modeling the Length of a Bird’s Wing
Example The table lists the weight W and the
wingspan L for birds of a particular species.
Use power regression to model the data with
L = aWb. Graph the data and the equation.
(b) Approximate the wingspan for a bird weighing 3.2 kilograms.
W (in kilograms)
0.5
1.5
2.0
2.5
3.0
0.77
1.10
1.22
1.31
1.40
L (in meters)

8. 4.4 Modeling the Length of a Bird’s Wing
Solution
(a) Let x be the weight W and y be the length L. Enter the data, and then select power regression
(PwrReg), as shown in the following figures.

9. 4.4 Modeling the Length of a Bird’s Wing
The resulting equation and graph can be seen in the figures below.
(b) If a bird weighs 3.2 kg, this model predicts the wingspan to be

10. 4.4 Graphs of Root Functions: Even Roots

11. 4.4 Graphs of Root Functions: Odd Roots

12. 4.4 Finding Domains of Root Functions
Example Find the domain of each function.
(a) (b)
Solution
4x + 12 must be greater than or equal to 0 since the root, n = 2, is even.
(b) Since the root, n = 3, is odd, the domain of g is all real numbers.
The domain of f is [–3,).

13. 4.4 Transforming Graphs of Root Functions
Example Explain how the graph of
can be obtained from the graph of
Solution
Shift left 3 units and stretch vertically by a factor of 2.

14. 4.4 Transforming Graphs of Root Functions
Example Explain how the graph of
can be obtained from the graph of
Solution
Shift right 1 unit, stretch vertically by a factor of 2, and reflect across the x-axis.

15. 4.4 Graphing Circles Using Root Functions
The equation of a circle centered at the origin with radius r is found by finding the distance from the origin to a point (x,y) on the circle.
The circle is not a function, so imagine a semicircle on top and another on the bottom.

16. 4.4 Graphing Circles Using Root Functions
Solve for y:
Since y2 = –y1, the “bottom” semicircle is a reflection of the “top” semicircle.

17. 4.4 Graphing a Circle Example Use a calculator in function mode to
graph the circle
Solution This graph can be obtained by graphing
in the same
window.
Technology Note: Graphs may not connect when using a non-decimal window.