,MAT1514 Assignment 1 (COMPLETE ANSWERS)
2025 - DUE 2025; 100% TRUSTED Complete,
trusted solutions and explanations.
MULTIPLE CHOICE,ASSURED EXCELLENCE
Question 1 Given f (x) = 3x2 − 4x + 7 and g(x) = x2 + 1. Find and
simplify the following: apm1514 1. (g f )(x) (4) 2. (g − f )(x) (2) 3.
g f (x) (4) 4. g−1(x) (3) [13 marks] 2 Downloaded by Corona Virus
() lOMoARcPSD| MAT1514/101/0/2025 Question 2 1. Use the
graph shown below to determine the intervals on which the
function is increasing, decreasing, or constant. (3) 2. Determine
whether the following relations are functions or not: (a) {(1, 2),
(3,−1), (−2, 3), (1,−3)} (1)mat1511 mat1512 (b) y = 3(x + 2)2 − 5
(1) (c) x2 − y2 = 9 (1) 3. Determine whether the lines y = 5
mat1513 3x +2 and 7x −2y = 4 are parallel, perpendicular, or
neither. (4) [10 marks] Question 3 1. If f (x) = x + 2 x − 3 ,
evaluate f (−2i ). (4) 2. Find the inverse of the function f (x) = 4x
+ 5. (5) 3. Given the following information about a polynomial
function, find the function: • The function has a zero of
multiplicity 2 at x = −1 and another zero at x = 4. mat1501
mat1503 • The function contains the point (2,−5). (5) 4. Find
the quotient and remainder if 4x3 + 7x2 − x + 2 is divided by 2x
− 1. (4) 5. Find the domain of f (x) = 3 5x − 4 and express your
answer in interval notation. (4) [22 marks] Total: 45 marks
Question 1
, 1. Find (g ∘ f)(x) (composition of functions g and f)
(g∘f)(x)=g(f(x))=g(3x2−4x+7)(g \circ f)(x) = g(f(x)) = g(3x^2 - 4x +
7)(g∘f)(x)=g(f(x))=g(3x2−4x+7)
Since g(x)=x2+1g(x) = x^2 + 1g(x)=x2+1, substitute f(x)f(x)f(x):
g(3x2−4x+7)=(3x2−4x+7)2+1g(3x^2 - 4x + 7) = (3x^2 - 4x + 7)^2
+ 1g(3x2−4x+7)=(3x2−4x+7)2+1
Expanding:
=9x4−24x3+49x2−56x+49+1= 9x^4 - 24x^3 + 49x^2 - 56x + 49 +
1=9x4−24x3+49x2−56x+49+1 =9x4−24x3+49x2−56x+50= 9x^4 -
24x^3 + 49x^2 - 56x + 50=9x4−24x3+49x2−56x+50
2. Find (g − f)(x) (difference of functions g and f)
(g−f)(x)=g(x)−f(x)(g - f)(x) = g(x) - f(x)(g−f)(x)=g(x)−f(x)
=(x2+1)−(3x2−4x+7)= (x^2 + 1) - (3x^2 - 4x +
7)=(x2+1)−(3x2−4x+7) =x2+1−3x2+4x−7= x^2 + 1 - 3x^2 + 4x -
7=x2+1−3x2+4x−7 =−2x2+4x−6= -2x^2 + 4x - 6=−2x2+4x−6
3. Find (g ⋅ f)(x) (product of functions g and f)
(g⋅f)(x)=g(x)⋅f(x)(g \cdot f)(x) = g(x) \cdot f(x)(g⋅f)(x)=g(x)⋅f(x)
=(x2+1)⋅(3x2−4x+7)= (x^2 + 1) \cdot (3x^2 - 4x +
7)=(x2+1)⋅(3x2−4x+7)
Expanding:
=x2(3x2−4x+7)+1(3x2−4x+7)= x^2(3x^2 - 4x + 7) + 1(3x^2 - 4x +
7)=x2(3x2−4x+7)+1(3x2−4x+7) =3x4−4x3+7x2+3x2−4x+7= 3x^4 -
4x^3 + 7x^2 + 3x^2 - 4x + 7=3x4−4x3+7x2+3x2−4x+7
2025 - DUE 2025; 100% TRUSTED Complete,
trusted solutions and explanations.
MULTIPLE CHOICE,ASSURED EXCELLENCE
Question 1 Given f (x) = 3x2 − 4x + 7 and g(x) = x2 + 1. Find and
simplify the following: apm1514 1. (g f )(x) (4) 2. (g − f )(x) (2) 3.
g f (x) (4) 4. g−1(x) (3) [13 marks] 2 Downloaded by Corona Virus
() lOMoARcPSD| MAT1514/101/0/2025 Question 2 1. Use the
graph shown below to determine the intervals on which the
function is increasing, decreasing, or constant. (3) 2. Determine
whether the following relations are functions or not: (a) {(1, 2),
(3,−1), (−2, 3), (1,−3)} (1)mat1511 mat1512 (b) y = 3(x + 2)2 − 5
(1) (c) x2 − y2 = 9 (1) 3. Determine whether the lines y = 5
mat1513 3x +2 and 7x −2y = 4 are parallel, perpendicular, or
neither. (4) [10 marks] Question 3 1. If f (x) = x + 2 x − 3 ,
evaluate f (−2i ). (4) 2. Find the inverse of the function f (x) = 4x
+ 5. (5) 3. Given the following information about a polynomial
function, find the function: • The function has a zero of
multiplicity 2 at x = −1 and another zero at x = 4. mat1501
mat1503 • The function contains the point (2,−5). (5) 4. Find
the quotient and remainder if 4x3 + 7x2 − x + 2 is divided by 2x
− 1. (4) 5. Find the domain of f (x) = 3 5x − 4 and express your
answer in interval notation. (4) [22 marks] Total: 45 marks
Question 1
, 1. Find (g ∘ f)(x) (composition of functions g and f)
(g∘f)(x)=g(f(x))=g(3x2−4x+7)(g \circ f)(x) = g(f(x)) = g(3x^2 - 4x +
7)(g∘f)(x)=g(f(x))=g(3x2−4x+7)
Since g(x)=x2+1g(x) = x^2 + 1g(x)=x2+1, substitute f(x)f(x)f(x):
g(3x2−4x+7)=(3x2−4x+7)2+1g(3x^2 - 4x + 7) = (3x^2 - 4x + 7)^2
+ 1g(3x2−4x+7)=(3x2−4x+7)2+1
Expanding:
=9x4−24x3+49x2−56x+49+1= 9x^4 - 24x^3 + 49x^2 - 56x + 49 +
1=9x4−24x3+49x2−56x+49+1 =9x4−24x3+49x2−56x+50= 9x^4 -
24x^3 + 49x^2 - 56x + 50=9x4−24x3+49x2−56x+50
2. Find (g − f)(x) (difference of functions g and f)
(g−f)(x)=g(x)−f(x)(g - f)(x) = g(x) - f(x)(g−f)(x)=g(x)−f(x)
=(x2+1)−(3x2−4x+7)= (x^2 + 1) - (3x^2 - 4x +
7)=(x2+1)−(3x2−4x+7) =x2+1−3x2+4x−7= x^2 + 1 - 3x^2 + 4x -
7=x2+1−3x2+4x−7 =−2x2+4x−6= -2x^2 + 4x - 6=−2x2+4x−6
3. Find (g ⋅ f)(x) (product of functions g and f)
(g⋅f)(x)=g(x)⋅f(x)(g \cdot f)(x) = g(x) \cdot f(x)(g⋅f)(x)=g(x)⋅f(x)
=(x2+1)⋅(3x2−4x+7)= (x^2 + 1) \cdot (3x^2 - 4x +
7)=(x2+1)⋅(3x2−4x+7)
Expanding:
=x2(3x2−4x+7)+1(3x2−4x+7)= x^2(3x^2 - 4x + 7) + 1(3x^2 - 4x +
7)=x2(3x2−4x+7)+1(3x2−4x+7) =3x4−4x3+7x2+3x2−4x+7= 3x^4 -
4x^3 + 7x^2 + 3x^2 - 4x + 7=3x4−4x3+7x2+3x2−4x+7
