Section 4.4 Radical Functions

Objectives: 1. To restrict the domain of a function that is not one-to-one so that the inverse is a function. 2. To graph radical functions and give their domains.

Definition Radical function A radical function is a function containing at least one variable in the radicand.

The most common radical function is the square root function The most common radical function is the square root function. All even root functions have both a positive and a negative root, but only the principal (positive) root is used. A square root function, whose simplest form is f(x) = x indicates only the principal square root.

x y x f(x) = x 1 2 3 4 7 1 2  1.4 3  1.7 2 7  2.6

EXAMPLE 1 Find the function described by the rule h(x) = x + 3 where the domain is {0, -3, 1, 4, 6}.

EXAMPLE 1 Find the function described by the rule h(x) = x + 3 where the domain is {0, -3, 1, 4, 6}.

Rational functions require restrictions so the denominator is not zero Rational functions require restrictions so the denominator is not zero. Radical functions require the radicand to be nonnegative.

Find the domain of the following function. 2 x 3 6 ) ( g + = 2 x 3 ³ + 2 x 3 - ³ 3 2 x -  þ ý ü î í ì -  = 3 2 x D g

Find the domain of the following function. 3 4 x ) ( p - = Dp = all real numbers Since the index (3) is odd, no restrictions are needed.

EXAMPLE 2 Find the domain for each of the following functions. a. g(x) = 3 4 – 3x – 7, b. q(x) = 5x + 7, c. r(x) = 2x + 4 (x – 2) x - 1 3

EXAMPLE 2 Find the domain for each of the following functions. a. g(x) = 3 4 – 3x – 7 a. for g(x) 4 – 3x  0 -3x  -4 x  4 3 Dg = {x|x  } 4 3

EXAMPLE 2 Find the domain for each of the following functions. b. q(x) = 5x + 7 3 b. for Q(x) Dq = {all real numbers}

EXAMPLE 2 Find the domain for each of the following functions. c. r(x) = 2x + 4 (x – 2) x - 1 c. for r(x) x – 1  0, x – 2  0 x  1 x  2 Dr = {x|x  1 and x  2}

EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4 EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4. If the inverse is not a function, then restrict the domain so that it becomes one. y = x2 + 4 x = y2 + 4 y2 = x – 4 y = ± x – 4 f-1(x) = ± x – 4

EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4 EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4. If the inverse is not a function, then restrict the domain so that it becomes one. If the domain of f(x) is restricted to x  0 then f(x) would be one-to-one and the function would have an inverse.

EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4 EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4. If the inverse is not a function, then restrict the domain so that it becomes one. If only the blue portion of f is used, then f-1(x) = x – 4. The blue portion of f-1 is the inverse function.

EXAMPLE 4 Find the inverse function for f(x) = - x – 4. y = - x – 4 x = - y – 4 x2 = (- y – 4)2 x2 = y – 4 x2 + 4 = y f-1(x) = x2 + 4, x  0

Homework pp. 191-193

►A. Exercises Give the domain of each. Also identify the x-intercept and the y-intercept. 7. q(x) = x + 1

►A. Exercises Give the domain of each. Also identify the x-intercept and the y-intercept. 9. r(x) = x + 2 3

►A. Exercises Give the domain of each. Also identify the x-intercept and the y-intercept. 11. f(x) = 2x – 4 + 7.75 5

►B. Exercises Graph each radical function. Identify the domain and range. 13. g(x) = -x

►B. Exercises For each graph below, find a one-to-one portion of the graph (restrict the domain), and then sketch the inverse. Write the restricted domains of f(x) and f-1(x) in interval notation.

►B. Exercises 19.

►B. Exercises 21.

■ Cumulative Review Find the areas of the following triangles. 26. with sides of 29, 36, and 47.

■ Cumulative Review Find the areas of the following triangles. 27. with A = 37°, b = 12.5, and c = 17.0

■ Cumulative Review Find the areas of the following triangles. 28. with a leg of 25 and a hypotenuse of 47.

■ Cumulative Review 29. Which two trig functions have a period of ?

■ Cumulative Review 30. What type of symmetry does an odd function have?