# Find Range of Square Root Functions

Find the range of square root functions; examples and matched problems with their answers at the bottom of the page.

 Graphical Analysis of Range of Square Root Functions The range of a function y = f(x) is the set of values y takes for all values of x within the domain of f. What is the range of f(x) = √x? The domain of f above is the set of all values of x in the interval [0 , +∞) As x takes values from 0 to +∞, √x takes values from √0 to √+∞ (see blue graph below). Hence the range of √x is given by the interval: [0 , +∞) The range of function of the form √x + k, (see red and green graphs below) is given by the interval:[0 , +∞). Right and left shifts do not affect the range of function. We can also write the range y = f(x) = √x + k of in the following inequality form y ≥ 0       or       √x + k ≥ 0 Fig1. - Examples of Square Root Functions. Example 1: Find the range of function f defined by f(x) = √x - 1 Solution to Example 1 We know, from the discussion above, that the range of function f(x) = √x is given by the interval [0 , +∞). The graph of the given function f(x) = √x - 1 is the graph of √ x shifted 1 unit to the right. A shift to the right does not affect the range. Hence the range of f(x) = √x - 1 is also given by the interval: [ 0 , +∞) Matched Problem 1: Find the range of function f defined by f(x) = √x + 5 Example 2: Find the range of function f defined by f(x) = - √x + 2 Solution to Example 2 We first start with the range of values of expression √x + 2 which may be written in inequality form as follows √x + 2 ≥ 0 Multiply both sides of the inequality by -1 to obtain to obtain the inequality - √x + 2 ≤ 0 The range of the expression - √x + 2 which is also the range of the given function is given by the interval ( -∞ , 0] Matched Problem 2: Find the range of function f defined by f(x) = - √x - 4 Example 3: Find the range of function f defined by f(x) = - 2 √x + 3 + 5 Solution to Example 3 The range of values of √x + 3 may be written as an inequality √x + 3 ≥ 0 Multiply both sides by -2 to obtain - 2 √x + 3 ≤ 0 Add 5t both sides of the above inequality to obtain - 2 √x + 3 + 5 ≤ 5 The range of values of the expression on the left side of the inequality, which is also the range of the given function, is given by the interval ( - ∞ , 5 ] Matched Problem 3: Find the range of function f defined by f(x) = - 5 √x + 3 - 10 Example 4: Find the range of function f defined by f(x) = √16 - x2 Solution to Example 4 We first need to find the domain of the given function defined as the values of x such that 16 - x2 ≥ 0 The solution set to the above inequality is the domain of f(x) and is given by the interval [ -4 , 4 ] The range of values of 16 - x2 for x in the interval [ -4 , 4 ] (domain) is given by the interval [0 , 16] since the graph is a parabola with a maximum at the point (0 , 16). The given function is the square root of 16 - x2 and therefore has the range defined by the interval [ √ 0 , √ 16 ] = [ 0 , 4 ]. See graphs below for better understanding Fig2. - Range of Square Root Functions. Matched Problem 4: Find the range of function f defined by f(x) = √4 - x2 Example 5: Find the range of function f defined by f(x) = √x2 - 25 Solution to Example 5 The domain of the given function is the set of x values such that x2 - 25 ≥ 0 The solution set to the above inequality is the domain of f(x) and is given by the interval ( - ∞ , -5] ∪ [5 , + ∞) For x in the interval ( - ∞ , -5] ∪ [5 , + ∞) the range of the expression x2 - 25 is given by the interval [ 0 , +∞). The given function is the square root of x2 - 25. Hence the range of the given function is given by the interval [ √ 0 , √+∞) = [ 0 , +∞) . See graph below for better understanding. Fig3. - Range of Square Root Functions. Matched Problem 5: Find the range of function f defined by f(x) = √x2 - 1 Example 6: Find the range of function f defined by f(x) = √x2 - 4x + 8 Solution to Example 6 The domain of the given function is the set of x values such that x2 - 4x + 8 ≥ 0 The discriminant of the quadratic expression x2 - 4x + 8 is given by (-4)2 - 4 (1)(8) = -16 Since the discriminant is negative, the quadratic expression is either positive or negative for all values of x. A test with x = 0 reveals the expression x2 - 4x + 8 is always positive and therefore the domain of the given function is the set of all real numbers. We next find the range of the expression x2 - 4x + 8 which may be written as x2 - 4x + 8 = (x - 2)2 + 4 The graphs of (x - 2)2 + 4 is a parabola with a minimum at (2 , 4) (the vertex). Hence the range of x2 - 4x + 8 is given by the interval [ 4 , +∞). The given function is the square root of x2 - 4x + 8 and therefore has the range given by [ √ 4 , √+∞ ) = [ 2 , +∞). See graph below for better understanding. Fig4. - Range of Square Root Functions. Matched Problem 6: Find the range of function f defined by f(x) = √x2 + 2x + 10 Answers for Matched Problems 1)       [0 , +∞) 2)       ( -∞ , 0 ] 3)       ( -∞ , -10 ] 4)       [ 0 , 2 ] 5)       [ 0 , +∞) 6)       [ 3 , +∞) More Find domain and range of functions, Find the range of functions, find the domain of a function and mathematics tutorials and problems.