Exponential Function and Its Graph: PECTAA Class 12 Maths, Punjab Board

Fold a paper again and again: 1, 2, 4, 8 layers. Or share a sweet again and again: ½, ¼, ⅛. Both are exponential. In Unit 1 of the PECTAA Mathematics 12 book you learn to read such a function from a short list of facts, and then sketch its graph without plotting many points.

The unit is called Graphical Representation of Functions. Section 1.4 covers the exponential function. This note is for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Definition. The function f(x) = aˣ for all real x, with a > 0 and a ≠ 1, is an exponential function. The number a is the base. The function f(x) = eˣ is the natural exponential function. The book defines e by an infinite series and gives its approximate value as 2.71828.

Growth and decay. The book gives eˣ and 2ˣ as exponential growth. It gives (½)ˣ as exponential decay.

Where it sits among functions (section 1.3). The book calls a function algebraic if it is obtained from x by a finite number of additions, subtractions, multiplications, divisions and nth roots. A function that is not algebraic is transcendental. Examples in the book are eˣ, 2ˣ, ln x, sin x and sin⁻¹x. The fundamental transcendental functions are y = aˣ (a > 0, a ≠ 1), ln x, sin x and sin⁻¹x. Combinations such as sin x + ln x, or x sin x, or ln 2x are non-fundamental.

Key points

The book studies y = aˣ (a > 0, a ≠ 1) by five properties.

1. Domain. Any real number can replace x, so the domain is ]−∞, ∞[. The graph has no jump or hole, because lim aˣ = aᶜ as x → c, so it is continuous everywhere.
2. Positivity and range. aˣ > 0 for every real x. The book shows this for positive integers (a product of positive numbers), for negative integers (the reciprocal of a positive number), for 0 (a⁰ = 1) and for rational powers. For irrational x it accepts the fact and says a rigorous proof is beyond this level. So the graph lies above the x-axis and the range is ]0, ∞[.
3. Intercepts. Put x = 0 to get y = a⁰ = 1. So the y-intercept is (0, 1). Put y = 0 and you need aˣ = 0, which never happens. So there is no x-intercept.
4. Increasing or decreasing. For x₁ < x₂ we have x₂ − x₁ > 0, and a^(x₂) = a^(x₁) · a^(x₂−x₁). If a > 1, then a^(x₂−x₁) > 1, so a^(x₂) > a^(x₁) and the function is strictly increasing. If 0 < a < 1, then a^(x₂−x₁) < 1, so the function is strictly decreasing.
5. Injective and asymptote. Because it is strictly increasing or strictly decreasing, the function is one-to-one. Every horizontal line meets the graph at most once. If a > 1, aˣ → 0 as x → −∞. If 0 < a < 1, aˣ → 0 as x → ∞. So y = 0 is the horizontal asymptote.

Book note on asymptotes. A horizontal asymptote is a line y = c that the function approaches when x becomes very large or very small. The graph does not have to touch or cross it.

Worked examples

Example 1 (book Example 7). Describe f(x) = eˣ.

1. Domain ]−∞, ∞[. Range ]0, ∞[.
2. The graph cuts the y-axis at (0, 1).
3. y = 0 is the horizontal asymptote.
4. It is continuous and strictly increasing, because e > 1.
5. Some values: e¹ ≈ 2.718, e² ≈ 7.389, e⁻¹ ≈ 0.368 and e⁻² ≈ 0.135.

Example 2 (book Example 8). Describe f(x) = (½)ˣ.

1. Domain ]−∞, ∞[. Range ]0, ∞[.
2. The graph cuts the y-axis at (0, 1).
3. y = 0 is the horizontal asymptote.
4. It is continuous and strictly decreasing, because the base ½ is less than 1.
5. Values: f(1) = 0.5, f(2) = 0.25, f(3) = 0.125, f(−1) = 2, f(−2) = 4 and f(−3) = 8.

Example 3. Is f(x) = 3ˣ increasing or decreasing? Give its y-intercept.

1. The base is 3, which is greater than 1.
2. So f is strictly increasing.
3. The y-intercept is (0, 1), since 3⁰ = 1.

Example 4. Is the graph of y = 2ˣ ever below the x-axis?

1. For a = 2, aˣ > 0 for every real x.
2. So the graph always lies above the x-axis.

Common mistakes in the exam

1. Saying the range is all real numbers. The range is ]0, ∞[.
2. Giving an x-intercept. There is none.
3. Writing the y-intercept as (1, 0). It is (0, 1).
4. Mixing up growth and decay. Base greater than 1 gives growth. Base between 0 and 1 gives decay.
5. Allowing a = 1 or a ≤ 0. The definition needs a > 0 and a ≠ 1.
6. Drawing the graph through the x-axis. It approaches y = 0 but does not cross it.

Exam technique

• Write the five properties in the book's order: domain, range, intercepts, increasing or decreasing, asymptote.
• Mark (0, 1) and two or three other points on your sketch, as the book does.
• Draw the curve smooth, and draw the asymptote as a dotted line.
• Check your board's notice for the sections in your paper.

Practice questions

1. State the domain and range of y = 5ˣ. Answer: Domain ]−∞, ∞[. Range ]0, ∞[.
2. Where does y = (1/3)ˣ cut the y-axis, and is it increasing? Answer: At (0, 1). It is strictly decreasing, since 0 < 1/3 < 1.
3. Name the horizontal asymptote of y = eˣ. Answer: y = 0.
4. Find f(−1) and f(2) for f(x) = (½)ˣ. Answer: f(−1) = 2 and f(2) = 0.25.
5. Is y = 2ˣ + ln x a fundamental transcendental function? Answer: No. It combines two transcendental functions, so the book calls it non-fundamental.

Ready to practise this? Continue in 2nd Year Mathematics.