Test Session Chapter 2 Functions and Graphs (1st Year Mathematics)

Multiple Choice Questions (MCQs)

Short Answer Questions

Important Concepts

Function Notation:

f(x) = expression in x

Domain: Set of all possible x-values

Range: Set of all possible y-values

Linear Function:

y = mx + b

m = slope, b = y-intercept

Difference Quotient:

[f(a+h) – f(a)] / h

Represents average rate of change

Exponential Function:

f(x) = a·e^(kx)

Growth: k > 0, Decay: k < 0

Function Composition:

(f∘g)(x) = f(g(x))

Apply g first, then f to the result

Function Calculator

Function f(x):

Value of x:

Slope Calculator

Point 1 (x₁, y₁):

Point 2 (x₂, y₂):

Detailed Answer Questions

Question 1 (5 marks)

Let g: R → R be given by g(x) = x³ – 3x. Determine if g(x) is injective and/or surjective.

Question 2 (5 marks)

A building’s height over time is modeled by H(t) = 100 + 20t (meters, t in months). The height of a growing tree nearby is given by T(t) = 50 + 10t + t². (i) At which time will the building and tree have the same height? (ii) What will that height be? Sketch the graphs and determine when the tree overtakes the building.

Question 3 (5 marks)

Consider the functions: f(x) = 2x + 3, g(x) = x² (i) Find f(g(x)) and g(f(x)) (ii) Find the inverse of f(x) (iii) Prove that f(f⁻¹(x)) = x (iv) Determine domain and range of both functions

Note: Answer any 2 questions. Each question carries 5 marks.

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