This article covers the essential skills of exponential functions and their graphs – a key topic in DSE Paper 1 Section A(2) and Section B worth 4–6 marks. You will learn the definition of exponential functions,their properties (domain,range,monotonicity,and asymptotes),how to sketch exponential graphs,and how to apply transformations. The article also covers the laws of exponents and introduces exponential growth and decay models. Includes step-by-step worked examples,DSE exam techniques,practice questions. These are essential skills that appear regularly in DSE papers.
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Exponential Functions | Definition, Properties & Graphs
Question 1 MC
Find the domain of \( f(x) = 5^x \).
A. \( x > 0 \) B. \( x \ge 0 \) C. All real numbers D. \( x \neq 0 \)
Question 2 MC
Which of the following is an increasing exponential function?
A. \( y = (0.5)^x \) B. \( y = 2^x \) C. \( y = \left(\frac{1}{3}\right)^x \) D. \( y = 0.9^x \)
Question 3 Short Answer
Simplify \( (2^3 \times 2^5) \div 2^4 \).
Question 4 Short Answer
Sketch \( y = 3^x + 2 \), labeling the asymptote and intercept.
Question 5 MC
Solve \( 2^{x-1} = 16 \).
A. \( 3 \) B. \( 4 \) C. \( 5 \) D. \( 6 \)
Question 6 Short Answer
A population of 500 decreases by 10% each year. Write an exponential model and find the population after 4 years.
Question 7 MC
What is the asymptote of \( y = 2^{-x} - 1 \)?
A. \( y = 0 \) B. \( y = -1 \) C. \( y = 1 \) D. \( x = 0 \)
Question 8 Short Answer
Describe the transformation from \( y = 2^x \) to \( y = -2^{x+1} \).
Question 9 Short Answer
Solve \( 3^{2x} = 27 \).
Question 10 Short Answer
Find the y-intercept of \( f(x) = 4^{x-1} + 2 \).
When solving exponential equations, make sure both sides have the same base before equating exponents. If not, rewrite using powers (e.g., \( 8 = 2^3 \)).