Key rules f f f is one-to-one if f ( x 1 ) = f ( x 2 ) ⟹ x 1 = x 2 f(x_1) = f(x_2) \implies x_1 = x_2 f ( x 1 ) = f ( x 2 ) ⟹ x 1 = x 2 (equivalently x 1 ≠ x 2 ⟹ f ( x 1 ) ≠ f ( x 2 ) x_1 \neq x_2 \implies f(x_1) \neq f(x_2) x 1 = x 2 ⟹ f ( x 1 ) = f ( x 2 ) ). Graphically: the Horizontal Line Test. To show f f f is NOT one-to-one, give two different inputs with the same output (e.g. f ( 1 ) = f ( − 1 ) f(1) = f(-1) f ( 1 ) = f ( − 1 ) ). Theorem 1.3.1: a linear function f ( x ) = a x + b f(x) = ax + b f ( x ) = a x + b is one-to-one if and only if a ≠ 0 a \neq 0 a = 0 . To find f − 1 f^{-1} f − 1 : write y = f ( x ) y = f(x) y = f ( x ) , interchange x x x and y y y , solve for y y y , rename y y y as f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . f f f and g g g are inverses when f ( g ( x ) ) = x f(g(x)) = x f ( g ( x )) = x and g ( f ( x ) ) = x g(f(x)) = x g ( f ( x )) = x . Also D f − 1 = R f D_{f^{-1}} = R_f D f − 1 = R f and R f − 1 = D f R_{f^{-1}} = D_f R f − 1 = D f ; the graphs are reflections across y = x y = x y = x . Visual guide Graph of f(x) = x² for x ≥ 0 and its inverse f⁻¹(x) = √x, mirrored across the dashed line y = x. The point (2, 4) on f matches the point (4, 2) on the inverse. 1 2 3 4 5 1 2 3 4 5 (2, 4) (4, 2) f(x) = x², x ≥ 0 f⁻¹(x) = √x y = x A function and its inverse are mirror images across the dashed line y = x. Here f(x) = x² with x ≥ 0 and f⁻¹(x) = √x. The point (2, 4) on f becomes (4, 2) on f⁻¹.
Questions 55 total 1 Book example Example 3 Find the inverse function of f ( x ) = 2 x + 3 f(x) = 2x + 3 f ( x ) = 2 x + 3 . 2 Related problem Related Problem 1, part 1 Determine whether f ( x ) = x 3 − 3 x f(x) = x^3 - 3x f ( x ) = x 3 − 3 x is one-to-one. 3 Related problem Related Problem 1, part 2 Determine whether f ( x ) = 1 x 2 + 1 f(x) = \frac{1}{x^2 + 1} f ( x ) = x 2 + 1 1 is one-to-one. 4 Related problem Related Problem 1, part 3 Determine whether f ( x ) = x − 1 f(x) = \sqrt{x - 1} f ( x ) = x − 1 is one-to-one. 5 Related problem Related Problem 1, part 4 Determine whether f ( x ) = x 2 + 1 ( x ≥ 0 ) f(x) = x^2 + 1 \ (x \ge 0) f ( x ) = x 2 + 1 ( x ≥ 0 ) is one-to-one. 6 Book example Example 4, part 1 Find the inverse function of the one-to-one function f ( x ) = ( x − 1 ) 3 f(x) = (x - 1)^3 f ( x ) = ( x − 1 ) 3 . 7 Book example Example 4, part 2 Find the inverse function of the one-to-one function f ( x ) = x + 1 x − 2 , x ≠ 2 f(x) = \frac{x + 1}{x - 2}, \ x \neq 2 f ( x ) = x − 2 x + 1 , x = 2 . 8 Book exercise Required Exercises 1.3, #1 Use the Horizontal Line Test to determine whether the function in the graph is one-to-one. 9 Book exercise Required Exercises 1.3, #5 Use the Horizontal Line Test to determine whether the function in the graph is one-to-one. 10 Book exercise Required Exercises 1.3, #7 Determine whether f ( x ) = 2 x − 3 f(x) = 2x - 3 f ( x ) = 2 x − 3 is one-to-one. If it is one-to-one, find its inverse. 11 Book exercise Required Exercises 1.3, #8 Determine whether f ( x ) = x 2 − 4 f(x) = x^2 - 4 f ( x ) = x 2 − 4 is one-to-one. If it is one-to-one, find its inverse. 12 Book exercise Required Exercises 1.3, #11 Determine whether f ( x ) = x − 2 f(x) = \sqrt{x - 2} f ( x ) = x − 2 is one-to-one. If it is one-to-one, find its inverse. 13 Book exercise Required Exercises 1.3, #13 Determine whether f ( x ) = x − 1 x + 2 , x ≠ − 2 f(x) = \frac{x - 1}{x + 2}, \ x \neq -2 f ( x ) = x + 2 x − 1 , x = − 2 is one-to-one. If it is one-to-one, find its inverse. 14 Book exercise Required Exercises 1.3, #23 Show that f ( x ) = 3 x − 4 f(x) = 3x - 4 f ( x ) = 3 x − 4 and g ( x ) = x + 4 3 g(x) = \frac{x + 4}{3} g ( x ) = 3 x + 4 are inverse functions of each other. 15 Book exercise Required Exercises 1.3, #25 Show that f ( x ) = x 3 + 1 f(x) = x^3 + 1 f ( x ) = x 3 + 1 and g ( x ) = x − 1 3 g(x) = \sqrt[3]{x - 1} g ( x ) = 3 x − 1 are inverse functions of each other. 16 Book exercise Required Exercises 1.3, #31 Find the inverse of the function f ( x ) = 2 x − 2 5 f(x) = \frac{2x - 2}{5} f ( x ) = 5 2 x − 2 . 17 Book exercise Required Exercises 1.3, #35 Find the inverse of the function f ( x ) = x + 1 x − 3 , x ≠ 3 f(x) = \frac{x + 1}{x - 3}, \ x \neq 3 f ( x ) = x − 3 x + 1 , x = 3 . 18 Book exercise Required Exercises 1.3, #45 Find the inverse of the function f ( x ) = x + 1 − x − 1 , x ≥ 1 f(x) = \sqrt{x + 1} - \sqrt{x - 1}, \ x \ge 1 f ( x ) = x + 1 − x − 1 , x ≥ 1 . 19 Book exercise Problem 2 (page after Exercises) Let f ( x ) = x + x , x ≥ 0 f(x) = \sqrt{x + \sqrt{x}}, \ x \ge 0 f ( x ) = x + x , x ≥ 0 . (a) Show that f f f is one-to-one. (b) Find f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . (c) Determine the domain and range of both f f f and f − 1 f^{-1} f − 1 . 20 Self test MCQ Self Learning Test #1 Which of the following functions is one-to-one on R \mathbb{R} R ? 21 Self test MCQ Self Learning Test #2 The inverse of f ( x ) = 4 x − 1 f(x) = 4x - 1 f ( x ) = 4 x − 1 is: 22 Self test MCQ Self Learning Test #3 Which function is NOT one-to-one? 23 Self test MCQ Self Learning Test #4 If f ( x ) = x 2 + 4 x + 4 f(x) = \sqrt{x^2 + 4x + 4} f ( x ) = x 2 + 4 x + 4 , then f f f is one-to-one on: 24 Self test MCQ Self Learning Test #5 Let f ( x ) = x − 1 f(x) = \sqrt{x - 1} f ( x ) = x − 1 . Its inverse is: 25 Self test MCQ Self Learning Test #6 If f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 with domain x ≥ 0 x \ge 0 x ≥ 0 , then f − 1 ( x ) = f^{-1}(x) = f − 1 ( x ) = 26 Self test MCQ Self Learning Test #7 If f ( x ) = 1 x − 1 f(x) = \frac{1}{x - 1} f ( x ) = x − 1 1 , then f − 1 ( x ) = f^{-1}(x) = f − 1 ( x ) = 27 Self test MCQ Self Learning Test #8 Which statement is TRUE? 28 Self test MCQ Self Learning Test #9 If f ( x ) = x 3 − 1 f(x) = x^3 - 1 f ( x ) = x 3 − 1 and g ( x ) = x + 1 3 g(x) = \sqrt[3]{x + 1} g ( x ) = 3 x + 1 , then: 29 Self test MCQ Self Learning Test #10 Which function has NO inverse on R \mathbb{R} R ? 30 Tutorial Tutorial 3, Q1 1 Determine whether f ( x ) = 3 x − 2 5 f(x) = \frac{3x - 2}{5} f ( x ) = 5 3 x − 2 is (1-1). If it is (1-1), find its inverse. 31 Tutorial Tutorial 3, Q1 2 Determine whether f ( x ) = x 4 + 2 x 2 f(x) = x^4 + 2x^2 f ( x ) = x 4 + 2 x 2 is (1-1). If it is (1-1), find its inverse. 32 Tutorial Tutorial 3, Q1 3 Determine whether f ( x ) = x − 1 x + 2 , x ≠ − 2 f(x) = \frac{x - 1}{x + 2}, \ x \neq -2 f ( x ) = x + 2 x − 1 , x = − 2 is (1-1). If it is (1-1), find its inverse. 33 Tutorial Tutorial 3, Q1 4 Determine whether f ( x ) = x 2 + 4 x + 4 f(x) = \sqrt{x^2 + 4x + 4} f ( x ) = x 2 + 4 x + 4 , for x ≤ − 2 x \le -2 x ≤ − 2 , is (1-1). If it is (1-1), find its inverse. 34 Tutorial Tutorial 3, Q2 A Show that f ( x ) = 1 x + 3 f(x) = \frac{1}{x + 3} f ( x ) = x + 3 1 and g ( x ) = 1 x − 3 g(x) = \frac{1}{x} - 3 g ( x ) = x 1 − 3 are inverse functions of each other. 35 Tutorial Tutorial 3, Q2 B1 Let f f f be a one-to-one function defined by f ( x ) = 1 + x + 4 f(x) = 1 + \sqrt{x + 4} f ( x ) = 1 + x + 4 . Find f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . 36 Tutorial Tutorial 3, Q2 B2 Let f ( x ) = 1 + x + 4 f(x) = 1 + \sqrt{x + 4} f ( x ) = 1 + x + 4 . Determine the domain and range of f ( x ) f(x) f ( x ) . 37 Past midterm Midterm 1446H S1, Q2 C Given that f ( x ) = 1 − x 5 7 f(x) = \sqrt[7]{1 - x^5} f ( x ) = 7 1 − x 5 is a (1-1) function, find f − 1 f^{-1} f − 1 . 38 Past midterm Alternative Midterm 1446H S1, Q2 B Determine algebraically whether the function f ( x ) = x 2 + 1 f(x) = x^2 + 1 f ( x ) = x 2 + 1 , x ≥ − 5 x \ge -5 x ≥ − 5 is a (1-1) function or not. 39 Past midterm Alternative Midterm 1446H S1, Q2 C Given that f ( x ) = 1 x + 5 f(x) = \frac{1}{x + 5} f ( x ) = x + 5 1 is a (1-1) function, find f − 1 f^{-1} f − 1 . 40 Past midterm Midterm 1446H (photo), Q2 B Given that f ( x ) = 2 x + 3 4 x − 1 3 f(x) = \sqrt[3]{\frac{2x + 3}{4x - 1}} f ( x ) = 3 4 x − 1 2 x + 3 is a (1-1) function, find: (i) f − 1 f^{-1} f − 1 , (ii) the range of f − 1 f^{-1} f − 1 . 41 Question bank Midterm 1443H S1, Q2 B Determine whether f ( x ) = ∣ x ∣ f(x) = |x| f ( x ) = ∣ x ∣ is a 1-1 function or not. 42 Question bank Midterm 1443H S1, Q2 C Given that f ( x ) = 3 x + 2 f(x) = 3x + 2 f ( x ) = 3 x + 2 is a 1-1 function, find f − 1 f^{-1} f − 1 . 43 Question bank Midterm 1442H S1, Q2 B Let f ( x ) = x − 1 x + 2 f(x) = \frac{x - 1}{x + 2} f ( x ) = x + 2 x − 1 . (i) Find D f D_f D f . (ii) Show that f f f is one-to-one. 44 Question bank Alternative Midterm 1443H S1, Q2 B Determine whether f ( x ) = x 3 f(x) = x^3 f ( x ) = x 3 is a 1-1 function or not. 45 Question bank Alternative Midterm 1443H S1, Q2 C Given that f ( x ) = 4 x + 5 3 x − 2 f(x) = \frac{4x + 5}{3x - 2} f ( x ) = 3 x − 2 4 x + 5 is a 1-1 function, find the range of f f f . 46 Question bank Midterm 1443H S2, Q2 A Show that f ( x ) = 2 x + 1 x − 1 f(x) = \frac{2x + 1}{x - 1} f ( x ) = x − 1 2 x + 1 is a one-to-one (1-1) function. 47 Question bank Midterm 1438H S1, Q2 B Given that f ( x ) = 1 − 2 x 3 x + 2 f(x) = \frac{1 - 2x}{3x + 2} f ( x ) = 3 x + 2 1 − 2 x is a one-to-one function, find f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . 48 Question bank Alternative Midterm 1438H S1, Q2 B The exam asks: show that f ( x ) = x 2 + x − 6 , x < 0 f(x) = x^2 + x - 6, \ x < 0 f ( x ) = x 2 + x − 6 , x < 0 is a one-to-one function. Is it? 49 Question bank Alternative Midterm 1438H S2, Q2 B Show that f ( x ) = x 2 − 4 x − 5 , x > 2 f(x) = x^2 - 4x - 5, \ x > 2 f ( x ) = x 2 − 4 x − 5 , x > 2 is a one-to-one function. 50 Question bank First Midterm 1439/1440H S1, Q3 Consider f ( x ) = 3 x − 2 2 x + 5 f(x) = \frac{3x - 2}{2x + 5} f ( x ) = 2 x + 5 3 x − 2 . (A) Show that f f f is one-to-one. (B) Find f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . (C) Find the range of f f f . 51 Question bank First Midterm 1441H S1, Q3 A Show that all linear functions are one-to-one. 52 Question bank First Midterm 1441H S1, Q3 B Given that f ( x ) = 5 x + 7 f(x) = 5x + 7 f ( x ) = 5 x + 7 is one-to-one: (i) find the inverse of f f f , (ii) find the range of f f f . 53 Question bank First Midterm 1439/1440H S2, Q3 Consider f ( x ) = x + 1 x 3 f(x) = \sqrt[3]{\frac{x + 1}{x}} f ( x ) = 3 x x + 1 . (A) Show that f f f is one-to-one. (B) Find f − 1 ( x ) f^{-1}(x) f − 1 ( x ) . (C) Find the range of f f f . 54 Question bank First Midterm 1441H S2, Q3 A Determine algebraically whether f ( x ) = x 2 + 6 x + 4 , x ≥ − 6 f(x) = x^2 + 6x + 4, \ x \ge -6 f ( x ) = x 2 + 6 x + 4 , x ≥ − 6 is one-to-one or not. 55 Question bank First Midterm 1441H S2, Q3 B Given that f ( x ) = 4 x + 2 f(x) = \frac{4}{x + 2} f ( x ) = x + 2 4 is one-to-one: (i) find the inverse of f f f , (ii) find the range of f f f .