MAT1514 Assignment 1 (COMPLETE ANSWERS) 2025 - DUE 2025

MAT1514 Assignment 1
(COMPLETE ANSWERS)
2025 - DUE 2025

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, Question 1
Given:
 f(x) = 3x² − 4x + 7
 g(x) = x² + 1
1. (g ∘ f)(x)
This represents the composition of g with f, meaning we substitute f(x) into g(x).
(g ∘ f)(x) = g(f(x)) = g(3x² − 4x + 7) = (3x² − 4x + 7)² + 1
Now, we need to expand (3x² − 4x + 7)²: (3x² − 4x + 7)² = (3x² − 4x + 7)(3x² − 4x + 7) = 3x²(3x²
− 4x + 7) − 4x(3x² − 4x + 7) + 7(3x² − 4x + 7) = 9x⁴ − 12x³ + 21x² − 12x³ + 16x² − 28x + 21x² −
28x + 49 = 9x⁴ − 24x³ + (21 + 16 + 21)x² + (−28 − 28)x + 49 = 9x⁴ − 24x³ + 58x² − 56x + 49
Now, add 1: (g ∘ f)(x) = 9x⁴ − 24x³ + 58x² − 56x + 49 + 1 = 9x⁴ − 24x³ + 58x² − 56x + 50
2. (g − f)(x)
This represents the subtraction of f(x) from g(x).
(g − f)(x) = g(x) − f(x) = (x² + 1) − (3x² − 4x + 7) = x² + 1 − 3x² + 4x − 7 = (x² − 3x²) + 4x + (1 −
7) = −2x² + 4x − 6
3. g/f (x)
This represents the division of g(x) by f(x).
g/f (x) = g(x) / f(x) = (x² + 1) / (3x² − 4x + 7)
This expression is already simplified as the numerator and denominator have no common factors
that can be easily factored out.
4. g⁻¹(x)
To find the inverse of g(x) = x² + 1, we follow these steps:
1. Replace g(x) with y: y = x² + 1
2. Swap x and y: x = y² + 1
3. Solve for y: x − 1 = y² y = ±√(x − 1)
4. Replace y with g⁻¹(x): g⁻¹(x) = ±√(x − 1)
Question 2
1. Intervals of Increasing, Decreasing, or Constant Function
Looking at the graph (which is not provided here, so I will provide a general explanation of how
to determine these intervals):