Square-root and Absolute Functions By Mr Porter Function V (2,-4) 4 (2, 4) X-axis Y-axis X-axis Y-axis y1 = 6 -2x 6 3 √6 Square-root and Absolute Functions By Mr Porter

Square-Root Function: The important feature of this function is that f(x) must be a ZERO or POSITVE value, otherwise the function is undefined in the real number system. We will at this stage of the course, restrict f(x) to be linear functions, such as f(x) = 2x + 3 or f(x) = 4 - x. The function, , is equivalent to half of a horizontal parabola and has similar graphing features. Example Consider the function: , , squaring both sides y2 = x – 5 , re-arrange to make x the subject. x = y2 + 5 , which is the form of a horizontal parabola. The important fact in this example is that: x – 5 ≥ 0, to have any real answer. This means, the Domain is x ≥ 5 for all x in R, and the Range is y ≥ 0 for all y in R.

Graphing : Example 2: Sketch Hence, state clearly the functions domain and range. Graphing : To sketch , start with y = f(x). Solution: Find domain first, solve 6 – 2x ≥ 0 Example 1: Sketch Hence, state clearly the functions domain and range. -2x ≥ -6 This mean the curve goes to the left. x ≤ 3 X-axis Y-axis -5 2.5 y1=2x-5 Sketch the graph of y = 2x - 5. Sketch y1 = 6 - 2x on the number plane. X-axis Y-axis y1 = 6 -2x 6 3 Now, take the square-root, top half of a parabola √6 Now, take the square-root, top half of a parabola left. Domain is found by solving 2x - 5 ≥ 0 Remember, down here y = 2x - 5 is negative! Domain: all x ≥ 2.5 in R Range: All y ≥ 0 in R Domain: all x ≤ 3 in R Range: All y ≥ 0 in R

Harder Examples: Sketch the graph of the following function and clear state the domain and range of each. Sol: Find domain first, solve x +1 ≥ 0 Sol: Find domain first, solve 2x - 4 ≥ 0 x +1 ≥ 0 ==> x ≥ -1 , curves to the right . 2x – 4 ≥ 0 ==> x ≥ 2 , curves to the right . Sketch y1 = 2x – 4 on the same number plane and y2 = 5. Sketch y1 = x + 1 on the same number plane and y2 = 2. Because of the (–) in-front of the square-root, the curve goes under. Now, sketch the square-root curve to the right, starting at x = -1. Slide the curve up 5! Slide the curve up 2! y = x+1 y = 2 -1 x y y2 = 5 y1 = 2x- 4 2 Domain: All x ≥ -1 in R Range: All y ≥ 2 in R Domain: All x ≥ 2 in R Range: All y ≤ 5 in R

Exercise: For each of the following: a) Sketch the function b) Define the domain and range of the function y -6 x -√6 4 2 y x -3 x -2 5 y Domain: all x ≤ 4 in R Range: all y ≥ 0 in R Domain: all x ≥ -6 in R Range: all y ≤ 0 in R Domain: all x ≥ -3 in R Range: all y ≥-2 in R

Absolute Value Functions. y = | f(x) | The absolute symbols | .. | given the value as a POSITIVE number, as in distance. Examples: Sketch and define domain and range. Step 1: Sketch f(x) = 2x – 3 -3 11/2 x y (2,-4) Step 1: Sketch f(x) = x2 - 4x 4 Step 2: at the x-intercept, reflect graph UP. 3 Step 2: at the x-intercepts, reflect negative part of graph UP. (2, 4) Domain: all x in R Range: all y ≥ 0 Domain: all x in R Range: all y ≥ 0

More Examples Domain: all x in R, x ≠ 1 Range: all y in R, y > 0 Step 1: Sketch f(x) = , asymptote at x = 1, y = 0 1 -2 Step 1: Sketch f(x) = , asymptote at x = 1, y = 0 1 2 Step 2: Reflect the negative part of the graph UP above the x-axis. 2 Step 2: Reflect the positive part of the graph DOWN below the x-axis. -2 [An absolute graph can be NEGATIVE] Domain: all x in R, x ≠ 1 Range: all y in R, y > 0 Domain: all x in R, x ≠ 1 Range: all y in R, y < 0

Exercise: For each function: i) Sketch the Function ii) Write down the domain and range. Domain: all x in R, x ≠ 1.5 Range: all y ≥ 0 in R, y ≠ 0 4 2 Domain: all x in R Range: all y ≥ 0 in R Note: f(x) = |(x - 2)(x + 2) | Note: f(x) = |(x - 6)(x + 2) | 2 -2 -4 4 Domain: all x in R Range: all y ≥ 0 in R -2 6 2 (2,-16) (2,16) Domain: all x in R Range: all y ≥ 0 in R