Chpt. 6, Radical Functions and Rational Exponents

Question 1

nth root

Answer 1

For any real numbers a and b, and any positive integer n, is a^n = b, then a is the nth root of b.

Question 2

principal root

Answer 2

When a number has two real roots, the positive root is called this. A radical sign indicates the principal root. the principle root of a negative number a is i∫ absolute value of a).

Question 3

radicand

Answer 3

The expression under the radical sign.

Question 4

index

Answer 4

With a radical sign, the index indicates the degree of the root. It is to the upper right of the radicand.

Question 5

the sign of the nth root

Answer 5

When the sign is even, there is both a primary and an alternate root. When it is odd, there is only the positive root.

Question 6

simplest form of a radical expression

Answer 6

The form in which there are no radicals in any denominator, no denominators in any radical, and no nth power factors in any radicand.

Question 7

rationalize the denominator

Answer 7

To rewrite it so that there are no radicals in any denominator, and no denominators in any radical. This is done by multiplying the entire expression by the denominator over the denominator, in fraction form. It doesn’t change the expression, because it’s equal to 1, but it does convert it to rational form.

Question 8

like radicals

Answer 8

Radical expressions that have the same index and the same radicand. Like radicals are added and subtracted using the distributive property. In other words, plus really means multiplication, and minus really means division.

Question 9

rational exponent

Answer 9

If the nth root of a is a real number and m is an integer, then a^(1/n) = 1/n∫a), where 1/n is the index. Likewise, a^(m/n) = n∫a)^m. If m is negative, then a does not equal zero.

Question 10

radical equation

Answer 10

A radical equation is an equation that has a variable in a radicand with a rational exponent.

Question 11

square root equation

Answer 11

A radical equation in which the radical has index 2.

Question 12

composite function

Answer 12

A combination of two functions such that the output for the first function becomes the input for the second function.

Only one function is represented with f, and the other with another letter, such as g.

Composite functions are not commutative.

Question 13

composite function operation signs:

1. addition
2. subtraction
3. multiplication
4. division
5. composition of functions

Answer 13

1. (f +g)(x) = f(x) + g(x)
2. (f -g)(x) = f(x) - g(x)
3. (f * g)(x) = f(x) * g(x)
4. (f / g)(x) = (f(x) / g(x), where g(x) cannot equal 0
5. f(g(x) = first evaluate g(x), then plug that solution into f(x)

Question 14

inverse relation

Answer 14

If a relation pairs element a of its domain with element b of its range, the inverse relation “undoes” the relation and pairs b with a. If (a, b) is an ordered pair of a relation, then (b, a) is an ordered pair of its inverse.

Question 15

inverse function

Answer 15

If function f pairs a value b with a, then its inverse, denoted f-superscript -1, pairs the value a with b. If the f-superscript -1 is also a function then f and f-superscript -1 are inverse functions.

Inverse functions set equal to each other will result in x = x when simplified. In other words f(f-superscript -1)(x) = x.

Question 16

one-to-one function

Answer 16

A function for which each y-value in the range corresponds to exactly one x-value in the domain. A one-to-one function f has an inverse f-superscript -1 that is also a function.

Question 17

radical function

Answer 17

A function that can be written in the form f(x) = a (index)n∫x -h) +k, where a does not equal zero. For even values of n, the domain of a radical function are the real numbers x, such that x ≥ h.

Question 18

square root function

Answer 18

A function that can be written in the form f(x) = a ∫x -h) +k, where a does not equal zero. The domain of a square root function is all real numbers x, such that x ≥ h.

Question 19

radical function transformation

from parent: y = (index)∫x)

1. reflection in x-axis
2. stretch
3. compress
4. horizontal translation
5. vertical translation
6. all combined

Answer 19

1. y = -(index)∫x)
2. y = -a(index)∫x), where a is > 1
3. y = -a(index)∫x), where a is

Question 20

square root function transformation

from parent: y = ∫x)

1. reflection in x-axis
2. stretch
3. compress
4. horizontal translation
5. vertical translation
6. all combined

Answer 20

1. y = -∫x)
2. y = a∫x), where a is > 1
3. y = a∫x), where a is < 1
4. y = ∫x -h)
5. y = ∫x) +k
6. y = -a∫x -h) +k