Key rules A function assigns exactly one output to each input. A relation from A A A to B B B is a function if every element of A A A appears exactly once as a first component. Domain: polynomials have domain R \mathbb{R} R ; rational functions exclude zeros of the denominator; even roots need a radicand ≥ 0 \ge 0 ≥ 0 ; odd roots allow every real number. Two functions are equal when they have the same domain and the same rule (e.g. x 2 = ∣ x ∣ \sqrt{x^2} = |x| x 2 = ∣ x ∣ , but x 2 − 9 x − 3 ≠ x + 3 \frac{x^2 - 9}{x - 3} \neq x + 3 x − 3 x 2 − 9 = x + 3 because the domains differ). D f ± g = D f g = D f ∩ D g D_{f \pm g} = D_{fg} = D_f \cap D_g D f ± g = D f g = D f ∩ D g , and D f / g = D f ∩ D g − { x : g ( x ) = 0 } D_{f/g} = D_f \cap D_g - \{x : g(x) = 0\} D f / g = D f ∩ D g − { x : g ( x ) = 0 } . ( f ∘ g ) ( x ) = f ( g ( x ) ) (f \circ g)(x) = f(g(x)) ( f ∘ g ) ( x ) = f ( g ( x )) ; its domain is every x x x in D g D_g D g with g ( x ) g(x) g ( x ) in D f D_f D f . Visual guide Graph of y = x. Line through the origin. Domain: all real numbers. y = x Graph of y = x². Parabola opening up. Domain: all real numbers. y = x² Graph of y = x³. Cubic through the origin. Domain: all real numbers. y = x³ Graph of y = √x. Square root starting at the origin. Domain: x ≥ 0. y = √x Graph of y = |x|. V shape with its corner at the origin. Domain: all real numbers. y = |x| Graph of y = 1/x. Two branches, not defined at x = 0. Domain: x ≠ 0. y = 1/x Six basic graphs and their domains. y = √x starts at 0, so its domain is x ≥ 0. y = 1/x has no point at x = 0, so its domain is x ≠ 0. The other four are defined for all real numbers.
Questions 53 total 1 Book example Example 4, part 1 Find the domain of f ( x ) = 3 x 4 − 2 x 2 + 7 f(x) = 3x^4 - 2x^2 + 7 f ( x ) = 3 x 4 − 2 x 2 + 7 . 2 Book example Example 4, part 2 Find the domain of g ( x ) = x + 1 x 2 − 9 g(x) = \frac{x + 1}{x^2 - 9} g ( x ) = x 2 − 9 x + 1 . 3 Book example Example 4, part 3 Find the domain of h ( x ) = 5 − 2 x h(x) = \sqrt{5 - 2x} h ( x ) = 5 − 2 x . 4 Book example Example 4, part 4 Find the domain of p ( x ) = { x + 1 , x < 0 1 x − 2 , x ≥ 0 p(x) = \begin{cases} \sqrt{x + 1}, & x < 0 \\ \frac{1}{x - 2}, & x \ge 0 \end{cases} p ( x ) = { x + 1 , x − 2 1 , x < 0 x ≥ 0 5 Related problem Related Problem 3, part 1 Determine whether the functions are equal: f ( x ) = 3 x − 1 f(x) = 3x - 1 f ( x ) = 3 x − 1 , g ( x ) = x + 2 x − 1 g(x) = x + 2x - 1 g ( x ) = x + 2 x − 1 6 Related problem Related Problem 3, part 2 Determine whether the functions are equal: f ( x ) = x 2 f(x) = \sqrt{x^2} f ( x ) = x 2 , g ( x ) = ∣ x ∣ g(x) = |x| g ( x ) = ∣ x ∣ 7 Related problem Related Problem 3, part 3 Determine whether the functions are equal: f ( x ) = x 2 − 9 x − 3 f(x) = \frac{x^2 - 9}{x - 3} f ( x ) = x − 3 x 2 − 9 , g ( x ) = x + 3 g(x) = x + 3 g ( x ) = x + 3 8 Related problem Related Problem 3, part 4 Determine whether the functions are equal: f ( x ) = x 2 − 4 x f(x) = \sqrt{x^2 - 4x} f ( x ) = x 2 − 4 x , g ( x ) = ∣ x − 2 ∣ g(x) = |x - 2| g ( x ) = ∣ x − 2∣ 9 Book example Example 9 Determine the specified combinations of the functions f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x , along with their corresponding domains: (1) ( f + g ) ( x ) (f + g)(x) ( f + g ) ( x ) , (2) ( f − g ) ( x ) (f - g)(x) ( f − g ) ( x ) , (3) ( f g ) ( x ) (fg)(x) ( f g ) ( x ) , (4) ( f g ) ( x ) \left(\frac{f}{g}\right)(x) ( g f ) ( x ) . 10 Book example Example 12, part 1 Let f ( x ) = 1 − x f(x) = \sqrt{1 - x} f ( x ) = 1 − x and g ( x ) = 1 2 − x g(x) = \frac{1}{2 - x} g ( x ) = 2 − x 1 . Determine ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) and its domain. 11 Book example Example 12, part 2 Let f ( x ) = 1 − x f(x) = \sqrt{1 - x} f ( x ) = 1 − x and g ( x ) = 1 2 − x g(x) = \frac{1}{2 - x} g ( x ) = 2 − x 1 . Determine ( g ∘ f ) ( x ) (g \circ f)(x) ( g ∘ f ) ( x ) and its domain. 12 Book exercise Required Exercises 1.2, #5 Let f f f be a relation from A = { 1 , 2 , 3 , 4 } A = \{1, 2, 3, 4\} A = { 1 , 2 , 3 , 4 } to B = { a , b , c , d } B = \{a, b, c, d\} B = { a , b , c , d } . Determine whether { ( 1 , a ) , ( 2 , b ) , ( 3 , b ) , ( 4 , c ) } \{(1, a), (2, b), (3, b), (4, c)\} {( 1 , a ) , ( 2 , b ) , ( 3 , b ) , ( 4 , c )} is a function. If it is, classify it as one-to-one or many-to-one. 13 Book exercise Required Exercises 1.2, #6 Let f f f be a relation from A = { 1 , 2 , 3 , 4 } A = \{1, 2, 3, 4\} A = { 1 , 2 , 3 , 4 } to B = { a , b , c , d } B = \{a, b, c, d\} B = { a , b , c , d } . Determine whether { ( a , 1 ) , ( b , 3 ) , ( c , 2 ) , ( d , 4 ) } \{(a, 1), (b, 3), (c, 2), (d, 4)\} {( a , 1 ) , ( b , 3 ) , ( c , 2 ) , ( d , 4 )} is a function. If it is, classify it as one-to-one or many-to-one. 14 Book exercise Required Exercises 1.2, #7 Let f f f be a relation from A = { 1 , 2 , 3 , 4 } A = \{1, 2, 3, 4\} A = { 1 , 2 , 3 , 4 } to B = { a , b , c , d } B = \{a, b, c, d\} B = { a , b , c , d } . Determine whether { ( 2 , a ) , ( 3 , d ) , ( 4 , c ) , ( 1 , a ) } \{(2, a), (3, d), (4, c), (1, a)\} {( 2 , a ) , ( 3 , d ) , ( 4 , c ) , ( 1 , a )} is a function. If it is, classify it as one-to-one or many-to-one. 15 Book exercise Required Exercises 1.2, #8 Let f f f be a relation from A = { 1 , 2 , 3 , 4 } A = \{1, 2, 3, 4\} A = { 1 , 2 , 3 , 4 } to B = { a , b , c , d } B = \{a, b, c, d\} B = { a , b , c , d } . Determine whether { ( 1 , b ) , ( 2 , c ) , ( 3 , d ) , ( 1 , a ) , ( 4 , c ) } \{(1, b), (2, c), (3, d), (1, a), (4, c)\} {( 1 , b ) , ( 2 , c ) , ( 3 , d ) , ( 1 , a ) , ( 4 , c )} is a function. If it is, classify it as one-to-one or many-to-one. 16 Book exercise Required Exercises 1.2, #24 Find the domain of f ( x ) = x − 8 3 f(x) = \sqrt[3]{x - 8} f ( x ) = 3 x − 8 . 17 Book exercise Communicate with Mathematics #1 A student says: "A relation is a function only if different inputs always give different outputs." Is the statement true or false? Justify. 18 Tutorial Tutorial 1+2, Q3 A1 Find the domain of f ( x ) = 3 + 2 x f(x) = \sqrt{3 + 2x} f ( x ) = 3 + 2 x . 19 Tutorial Tutorial 1+2, Q3 A2 Find the domain of f ( x ) = x + 1 x 2 − 4 f(x) = \frac{x + 1}{x^2 - 4} f ( x ) = x 2 − 4 x + 1 . 20 Tutorial Tutorial 1+2, Q3 B1 Let f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x . Find ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) and its domain. 21 Tutorial Tutorial 1+2, Q3 B2 Let f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x . Find ( g ∘ f ) ( x ) (g \circ f)(x) ( g ∘ f ) ( x ) and its domain. 22 Tutorial Tutorial 1+2, Q3 B3 Let f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x . Find ( f ∘ g ) ( 3 ) (f \circ g)(3) ( f ∘ g ) ( 3 ) . 23 Past midterm Midterm 1446H S1, Q2 A(i) Find the domain of f ( x ) = x − 5 3 − x f(x) = \sqrt{\frac{x - 5}{3 - x}} f ( x ) = 3 − x x − 5 . 24 Past midterm Midterm 1446H S1, Q2 A(ii) Find the domain of f ( x ) = 1 x − 3 x − 2 f(x) = \frac{1}{x} - \frac{3}{x - 2} f ( x ) = x 1 − x − 2 3 . 25 Past midterm Midterm 1446H S1, Q2 B Write the function h ( x ) = 1 ( x + 5 ) 7 h(x) = \frac{1}{(x + 5)^7} h ( x ) = ( x + 5 ) 7 1 as a composition of two functions. 26 Past midterm Alternative Midterm 1446H S1, Q2 A For f ( x ) = 2 x − 3 f(x) = \frac{2}{x - 3} f ( x ) = x − 3 2 and g ( x ) = 1 x + 2 g(x) = \frac{1}{x + 2} g ( x ) = x + 2 1 , find: (i) the formula of ( f g ) ( x ) \left(\frac{f}{g}\right)(x) ( g f ) ( x ) , and (ii) the domain of ( f g ) ( x ) \left(\frac{f}{g}\right)(x) ( g f ) ( x ) . 27 Past midterm Midterm 1446H (photo), Q2 A(i) Find the domain of f ( x ) = 1 x + tan x f(x) = \frac{1}{x} + \tan x f ( x ) = x 1 + tan x . 28 Past midterm Midterm 1446H (photo), Q2 A(ii) Find the domain of f ( x ) = 2 x − 6 ∣ x ∣ − 5 f(x) = \frac{\sqrt{2x - 6}}{|x| - 5} f ( x ) = ∣ x ∣ − 5 2 x − 6 . 29 Question bank Midterm 1443H S1, Q2 A Find the domain of f ( x ) = 3 x − 4 x + 3 f(x) = \sqrt{\frac{3x - 4}{x + 3}} f ( x ) = x + 3 3 x − 4 . 30 Question bank Midterm 1443H S1, Q2 D Let f ( x ) = x 2 − 1 4 f(x) = x^2 - \frac{1}{4} f ( x ) = x 2 − 4 1 and g ( x ) = 1 x g(x) = \frac{1}{x} g ( x ) = x 1 . Find the value of ( f ∘ g ) ( − 2 ) (f \circ g)(-2) ( f ∘ g ) ( − 2 ) . 31 Question bank Midterm 1442H S1, Q1 B(ii) Solve ( f ∘ g ) ( x ) ≥ − 8 (f \circ g)(x) \ge -8 ( f ∘ g ) ( x ) ≥ − 8 , where f ( x ) = 1 x f(x) = \frac{1}{x} f ( x ) = x 1 and g ( x ) = 1 x − 4 g(x) = \frac{1}{x - 4} g ( x ) = x − 4 1 . Write your answer in interval notation. 32 Question bank Midterm 1442H S1, Q2 A Find the domain of f ( x ) = 2 x 3 + 5 x − 3 f(x) = 2x^3 + 5x - 3 f ( x ) = 2 x 3 + 5 x − 3 . 33 Question bank Midterm 1442H S2, Q2 A Find the domain of f ( x ) = x − 5 f(x) = \sqrt{x - 5} f ( x ) = x − 5 . 34 Question bank Midterm 1442H S2, Q2 B Let f ( x ) = 6 x − 4 f(x) = \frac{6}{x - 4} f ( x ) = x − 4 6 and g ( x ) = x − 4 g(x) = x - 4 g ( x ) = x − 4 . Find: (i) f ⋅ g f \cdot g f ⋅ g , (ii) D f ⋅ g D_{f \cdot g} D f ⋅ g , (iii) ( f ∘ g ) ( 2 ) (f \circ g)(2) ( f ∘ g ) ( 2 ) . 35 Question bank Alternative Midterm 1443H S1, Q2 A Find the domain of f ( x ) = − x 2 + 2 x + 3 f(x) = \sqrt{-x^2 + 2x + 3} f ( x ) = − x 2 + 2 x + 3 . 36 Question bank Alternative Midterm 1443H S1, Q2 D Let f ( x ) = x 3 f(x) = x^3 f ( x ) = x 3 and g ( x ) = x 2 g(x) = x^2 g ( x ) = x 2 . Find the value of ( f ∘ g ) ( − 1 ) (f \circ g)(-1) ( f ∘ g ) ( − 1 ) . 37 Question bank Midterm 1443H S2, Q2 B Let f ( x ) = 1 x f(x) = \frac{1}{\sqrt{x}} f ( x ) = x 1 and g ( x ) = x 2 − 2 x + 1 g(x) = x^2 - 2x + 1 g ( x ) = x 2 − 2 x + 1 . Find: (i) ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) , (ii) D f ∘ g D_{f \circ g} D f ∘ g . 38 Question bank Midterm 1438H S1, Q2 A Let f ( x ) = 7 4 − x 2 f(x) = \frac{7}{4 - x^2} f ( x ) = 4 − x 2 7 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x . Find: (1) ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) , (2) D f D_f D f , D g D_g D g and D f ∘ g D_{f \circ g} D f ∘ g . 39 Question bank Alternative Midterm 1438H S1, Q2 A Let f ( x ) = 7 x − 2 f(x) = \frac{7}{\sqrt{x} - 2} f ( x ) = x − 2 7 and g ( x ) = x 2 + 2 g(x) = x^2 + 2 g ( x ) = x 2 + 2 . Find: (1) ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) , (2) D f D_f D f , D g D_g D g and D f ∘ g D_{f \circ g} D f ∘ g . 40 Question bank Midterm 1438H S2, Q1 B Find the domain of f ( x ) = x 2 − x − 6 f(x) = \sqrt{x^2 - x - 6} f ( x ) = x 2 − x − 6 . 41 Question bank Alternative Midterm 1438H S2, Q2 A Let f ( x ) = 3 x − 4 f(x) = \frac{3}{\sqrt{x - 4}} f ( x ) = x − 4 3 and g ( x ) = x 2 + 4 g(x) = x^2 + 4 g ( x ) = x 2 + 4 . Find: (1) ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) , (2) D f D_f D f , D g D_g D g and D f ∘ g D_{f \circ g} D f ∘ g . 42 Question bank First Midterm 1439/1440H S1, Q2 A Let f ( x ) = 4 − x 2 f(x) = \sqrt{4 - x^2} f ( x ) = 4 − x 2 and g ( x ) = x − 1 g(x) = \sqrt{x - 1} g ( x ) = x − 1 . Find: (1) ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) , (2) D f ∘ g D_{f \circ g} D f ∘ g . 43 Question bank First Midterm 1439/1440H S1, Q2 B Determine whether the functions f ( x ) = ( ∣ x ∣ + 4 ) ( x − 4 ) ∣ x ∣ + 4 f(x) = \frac{(|x| + 4)(x - 4)}{|x| + 4} f ( x ) = ∣ x ∣ + 4 ( ∣ x ∣ + 4 ) ( x − 4 ) and g ( x ) = x − 4 g(x) = x - 4 g ( x ) = x − 4 are the same or not. 44 Question bank First Midterm 1441H S1, Q2 A Find the domain of each: (i) h ( x ) = x 2 − 3 x − 4 h(x) = x^2 - 3x - 4 h ( x ) = x 2 − 3 x − 4 , (ii) g ( x ) = 5 x sin x g(x) = \frac{5x}{\sin x} g ( x ) = s i n x 5 x . 45 Question bank First Midterm 1441H S1, Q2 B Let f ( x ) = 1 x f(x) = \frac{1}{x} f ( x ) = x 1 and g ( x ) = 2 x − 1 g(x) = \frac{2}{x - 1} g ( x ) = x − 1 2 . Find: (i) the rule ( f + g ) ( x ) (f + g)(x) ( f + g ) ( x ) , (ii) D f + g D_{f + g} D f + g , (iii) the rule ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) . 46 Question bank First Midterm 1439/1440H S2, Q2 A Let f ( x ) = x f(x) = \sqrt{x} f ( x ) = x and g ( x ) = 9 − x 2 g(x) = \sqrt{9 - x^2} g ( x ) = 9 − x 2 . Find: (1) f + g f + g f + g , (2) D f + g D_{f + g} D f + g . 47 Question bank First Midterm 1441H S2, Q2 A Find the domain of each: (i) k ( x ) = x + 5 k(x) = \sqrt{x + 5} k ( x ) = x + 5 , (ii) h ( x ) = 2 x csc ( 4 x ) h(x) = \frac{2x}{\csc(4x)} h ( x ) = c s c ( 4 x ) 2 x . 48 Question bank First Midterm 1441H S2, Q2 B Let f ( x ) = cos x f(x) = \cos x f ( x ) = cos x and g ( x ) = x 2 + 5 g(x) = x^2 + 5 g ( x ) = x 2 + 5 . Find: (i) the rule ( f / g ) ( x ) (f/g)(x) ( f / g ) ( x ) , (ii) D f / g D_{f/g} D f / g , (iii) the rule ( g ∘ f ) ( x ) (g \circ f)(x) ( g ∘ f ) ( x ) . 49 AI generated MCQ Written to fill the MCQ gap The domain of f ( x ) = x − 3 x − 5 f(x) = \frac{\sqrt{x - 3}}{x - 5} f ( x ) = x − 5 x − 3 is … \ldots … 50 AI generated MCQ Written to fill the MCQ gap Let f ( x ) = x 2 + 1 f(x) = x^2 + 1 f ( x ) = x 2 + 1 and g ( x ) = x g(x) = \sqrt{x} g ( x ) = x . Then ( f ∘ g ) ( x ) (f \circ g)(x) ( f ∘ g ) ( x ) and its domain are … \ldots … 51 AI generated MCQ Written to fill the MCQ gap The domain of f ( x ) = x + 2 x 2 − x − 6 f(x) = \frac{x + 2}{x^2 - x - 6} f ( x ) = x 2 − x − 6 x + 2 is … \ldots … 52 AI generated MCQ Written to fill the MCQ gap Which pair of functions are equal? 53 AI generated MCQ Written to fill the MCQ gap If f ( x ) = 2 x − 1 f(x) = 2x - 1 f ( x ) = 2 x − 1 and g ( x ) = x 2 g(x) = x^2 g ( x ) = x 2 , then ( f ∘ g ) ( − 2 ) = … (f \circ g)(-2) = \ldots ( f ∘ g ) ( − 2 ) = …