5.4 Power and Radical Functions

5.4P ower and Radical Functions

This is part of 5.2 in the current text

In this section, you will learn about the properties and characteristics of power and radical functions. Upon completion you will be able to: • Identify if a given function is a power or a radical function. • Convert between radical and power notation. • Memorize the graphs of the parent even and odd power/radical functions (square root and cube root). • Determine the domain of a function involving power or radical functions, using interval notation. • Identify the conjugate of a given expression involving radicals. • Rationalize the numerator or denominator of a given expression, by multiplying by the conjugate.

Defining Power Functions A power function is a function with a single term that is the product of a real number and a variable raised to a ﬁxed real number.

Deﬁnition A power function is a function that can be represented in the form

f (x) = axp

where a and p are real numbers , and a is known as the coeﬃcient.

Here the left side tells us that x is the variable; a and p are parameters that can be any real numbers. Because a and p can be anything, this is a very general function type meaning that the properties of the function can be very diﬀerent based on those values of a and p. Before we dive into the properties of power functions, the authors would like to pause and review some basic properties of variables raised to numerical exponents.

Review of Exponent Laws Theorem 5.1 Laws of Exponents Let p and q be any real numbers and x and y be any variables with values greater than zero. Then, the following rules hold true.

xp xq = xp+q (xp)q = xpq 1 = x−p xp 1 = xp x−p 1 √ x p = p x (xy)p = xpyp !p x xp = y yp

! (x + y)p ,(xp + yp) and (x − y)p ,(xp − yp)

√ 1 Mathematics call p x the radical form and x p the exponent form. Both of these have the same meaning, they just √ 1 look a bit diﬀerent. Anytime you see p x, you can replace it with x p and vice versa.

These rules will all be quite handy in calculus. In both integral and diﬀerential calculus, we will have rules that work well when we have a power function, but won’t work for other forms of functions. By being able to rewrite 1 functions like f (x) = , as power function ( f (x) = x−2 here), other calculations will be simpliﬁed. x2 3  4  2 (xy)  Example1 Simplify   assuming x > 0 and y > 0.  1  xy 2

√ p 2 Example2 Simplify 2x + 3y assuming x > 0 and y > 0.

135 © TAMU Categories of Power Functions Returning to our discussion of power functions, we can categorize a power function, f (x) = axp by the value of its exponent (power). 1. If p = 0 and x , 0, then f (x) is the constant polynomial function, f (x) = ax0 = a. 2. If p is a positive integer, then f (x) is a pth degree polynomial function with only the leading term.

3. If p is a negative integer, then f (x) is a rational function. Each of these types of power functions hold the properties discussed in the Polynomial Functions and Rational Functions Sections, respectively. 4. If p is an irrational number and x > 0, then we have a complicated power function, which is beyond the scope of this text. 1 5. If p = , where n is a natural number, such that n ≥ 2, then f (x) is a radical (root) function. n

Describing Properties of Radical Functions Deﬁnition A radical (root) function is a function of the form

√n 1 f (x) = x = x n

where n is a natural number, such that n ≥ 2, called the index or root. • If n is an even number, then we say f (x) has an even root and f (x) is only deﬁned for x ≥ 0. (The even root of a negative number is not a real number.) • If n is an odd number, then we say f (x) has an odd root and f (x) is deﬁned for any real number x.

Name Function Graph Table Domain

x f (x) √ 0 0 Square root f (x) = x [0,∞) 1 1 4 2 -9-8-7-6-5-4-3-2-1123456789

Figure 5.4.1

x f (x) -1 -1 √ -0.125 -0.5 Cube root f (x) = 3 x (−∞,∞) 0 0 0.125 0.5 -9-8-7-6-5-4-3-2-1123456789 1 1 Figure 5.4.2

Table 5.1: A table with 3 rows and 5 columns. The ﬁrst row includes labels: Name, Function, Graph, Table, and Domain. The second row includes the corresponding properties for a square root parent function, while the third row include the corresponding properties for the cube root parent function.

Many times in Calculus we are faced with functions in radical form. It is often necessary to convert from radical form to an equivalent exponent (power) form.

Example3 Write the following radicals in their equivalent exponent (power) form. p a. R(p) = 15000 − 1.5p

r 3 1 −1 b. h(x) = x2 + 5x 2

1 Sometimes it is necessary to have a function written with non-negative powers using x−p = . xp p √ √ 2 2 2 N x = ( x) = |x|, not x, as x is non-negative, but if x is negative x is undeﬁned in the real numbers.

137 © TAMU As with all functions we need to investigate the domain of radical (root) functions. Our investigation will focus on the diﬀerences in domain based on whether the root is even or odd.

Example4 State the domain of each function, using interval notation. 2 5 a. f (y) = (y + 1) 4

4 b. h(z) = (6z + 1) 3

4x c. k(x) = √ 2x − 1

We ﬁnd the intercepts of power functions, including radicals, using the same techniques discussed in previous sections. As with rational functions, power functions may or may not have both x-intercepts and y-intercepts.

Computing Domains of Algebraic Functions By combining types of functions into a single function, we get, what the authors call, an algebraic function.

The domain of an algebraic function is all real numbers, x, which satisfy all parts of the function.So far our only discussed domain restrictions are 1. Division by zero is undeﬁned. 2. Even roots of negative numbers are undeﬁned.

Example5 State√ the domain of each algebraic function, using interval notation. a. f (x) = 3 − 18x − 15

4 − x2 b. h(x) = √ x − 6

√ 7 − x c. j(x) = 3x + 8

3 (5x − 9) 2 d. k(x) = √ 7 x + 2

139 © TAMU Rationalizing Numerators and Denominators We can remove radicals from the denominators of fractions using a process call rationalizing the denominator.

We know that multiplying by 1 does not change the value of the expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing the sum or diﬀerence of two terms where at least one term is a square root radical, multiply the numerator and denominator by the conjugate of the denominator, which√ is found by changing the sign between√ the two terms of the denominator.√ For example, if the denominator√ is a + b c, then the conjugate is a − b c, and if the denominator is a − b c, then the conjugate is a + b c. Similarly, we can rationalize a numerator containing radicals.

Example6 Rationalize the denominator and simplify the expression x − 36 √ for x ≥ 0 and x , 36 − x + 6

Example7 Find and simplify the diﬀerence quotient for the function √ r(x) = 2 x + 1

Reﬂection:

• Can you recognize a power/radical function and transition between the two forms of the same function? • Can you select the graphs of the parent power/radical functions from a group of graphs? • Can you explain how to ﬁnd the domain of a power/radical function based on the index? • Can you explain how a conjugate is used to rationalize an expression?