Exponential Functions
Suppose a colony of bacteria doubles every hour. After one hour you have twice what you started with; after two hours, four times; after three hours, eight times; after ten hours, more than a thousand times. The amount you have after t hours is described by the formula 2t, where the variable lives in the exponent. That is the defining feature of an exponential function — the variable is upstairs, not downstairs. This single structural difference is what gives exponentials their explosive growth and their distinctive curved shape.
This page introduces exponential functions: what they are, the two qualitatively different shapes they take depending on the base, what their graphs look like, and the special number e that appears throughout mathematics. The next pages then build on this foundation to introduce logarithms, which are the inverses of exponentials, and to solve equations involving both. Get this page solid and the rest of the topic falls into place.
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What an exponential function is
An exponential function is a function of the form f(x) = ax, where the base a is a positive constant not equal to 1, and the variable x sits in the exponent. The function 2x is exponential, and so is 10x, and (1/3)x. The function x² is not exponential, even though it involves an exponent — in x² the variable is the base and the exponent is fixed, which makes it a polynomial. The distinction is real and important: exponential functions and polynomial functions behave very differently as x grows.
Two restrictions on the base are worth understanding rather than just memorising. First, we require a > 0. The reason is that if a were negative, expressions like a1/2 would mean taking a square root of a negative number, which is undefined in the real numbers. We want our function to produce a real number for every real input x, including fractional inputs, so the base has to be positive. Second, we exclude a = 1, because 1x is just 1 for every x — it's a constant function, not genuinely exponential, and including it would only complicate the rules without adding anything useful.
The domain of an exponential function is all real numbers — you can raise a positive base to any power, positive or negative, integer or fractional. The range, however, is only the positive reals: ax > 0 for every x. This last fact is worth pausing on. Even when x is very negative, like x = −1000, the value a−1000 = 1/a1000 is a tiny positive number, not zero and certainly not negative. The function gets arbitrarily close to zero but never touches it. This is why the x-axis is called a horizontal asymptote of the graph.
The two regimes: growth and decay
The shape of the graph of f(x) = ax depends on whether the base a is greater than 1 or between 0 and 1. These two cases produce qualitatively different behaviour, and you should know both on sight.
When a > 1, the function grows as x increases. Each step to the right multiplies the value by a, so the function gets bigger and bigger — exponentially fast. Going left, each step divides by a, so the values shrink toward zero but stay positive. This is exponential growth. The function 2x is the standard example: at x = 0 it equals 1, at x = 1 it equals 2, at x = 10 it equals 1024, at x = 20 it exceeds a million.
When 0 < a < 1, the function does the opposite. As x increases, each step multiplies by a number less than 1, so the function shrinks. Going left, each step divides by a (multiplies by 1/a, which is greater than 1), so the function grows. This is exponential decay. The function (1/2)x is the standard example: at x = 0 it equals 1, at x = 1 it equals 1/2, at x = 10 it equals 1/1024, and as x → −∞ it grows without bound.
Notice three features the two graphs share. Both pass through the point (0, 1), because anything to the power 0 is 1. Both have the x-axis as a horizontal asymptote — one approaches it going left, the other going right, but neither curve ever touches the x-axis. And both are entirely above the x-axis, because ax is always positive when a is positive.
| Property | When a > 1 (growth) | When 0 < a < 1 (decay) |
|---|---|---|
| Behaviour | Increasing — grows as x increases | Decreasing — shrinks as x increases |
| As x → +∞ | y → +∞ | y → 0 |
| As x → −∞ | y → 0 | y → +∞ |
| y-intercept | (0, 1) | (0, 1) |
| Range | (0, ∞) | (0, ∞) |
| Examples | 2x, 10x, ex | (1/2)x, (1/3)x, e−x |
Evaluating exponentials
To evaluate an exponential function at a specific input, you raise the base to that power. The mechanical work is fluency with exponent rules — particularly negative exponents, fractional exponents, and the meaning of zero exponents — which you have likely seen before but which are worth reviewing here because they appear constantly in CSCA problems.
Three rules are used over and over. First, a0 = 1 for any positive a. So 50 = 1, 1000 = 1, even (1/7)0 = 1. Second, a negative exponent means a reciprocal: a−n = 1/an. So 2−3 = 1/23 = 1/8. Third, a fractional exponent means a root: a1/n = ⁿ√a, and am/n = (ⁿ√a)m. So 82/3 = (∛8)² = 2² = 4.
If f(x) = 3x, find f(−2).
Substitute x = −2: f(−2) = 3−2. By the negative-exponent rule, 3−2 = 1/32 = 1/9. So f(−2) = 1/9.
The reasoning behind this rule is worth keeping in mind. Going from 32 to 31 divides by 3 (from 9 to 3). Going from 31 to 30 divides by 3 again (from 3 to 1). Continuing the pattern, 3−1 divides 1 by 3 to give 1/3, and 3−2 divides again to give 1/9. The negative-exponent rule just continues the same divide-by-3 pattern to the left.
If f(x) = 16x, find f(3/4).
Substitute x = 3/4: f(3/4) = 163/4. By the fractional-exponent rule, 163/4 = (161/4)3. The fourth root of 16 is 2, because 24 = 16. Then 23 = 8. So f(3/4) = 8.
You could also compute this as (163)1/4 = 40961/4, but taking the root first usually gives smaller numbers and is easier to do by hand.
Transformations of exponential graphs
Once you know what y = ax looks like, you can read off graphs of related functions like y = ax + c or y = ax − h by shifting the basic curve. The transformations here follow the same rules as for any function — adding a constant outside shifts vertically, subtracting one inside shifts horizontally, multiplying by a constant outside stretches vertically — but it is worth seeing them once specifically for exponentials, because the asymptote moves with the graph and that catches students out.
The function y = ax + c is the basic curve shifted up by c units. The y-intercept moves from (0, 1) to (0, 1 + c), and crucially the horizontal asymptote moves from y = 0 to y = c. So y = 2x + 3 looks like the standard 2x curve raised by 3, with asymptote y = 3 instead of y = 0. The function y = ax − h shifts the basic curve right by h units (or left if h is negative). The asymptote stays at y = 0 in this case, but the y-intercept shifts.
The natural exponential and the number e
Among all possible bases, one in particular plays a starring role in mathematics: the number e ≈ 2.71828. The function ex is called the natural exponential function, and it shows up everywhere — in continuous compound interest, in radioactive decay, in population models, in the heart of calculus. The reason is technical (e is the unique base for which the exponential function is its own derivative), but for now, treat e as a particular constant just like π. It is roughly 2.7, it is bigger than 1, and so ex is an exponential growth function with all the standard properties.
You will see ex sometimes written as exp(x). They mean the same thing. Both notations are common and the CSCA exam can use either. The function ex goes through (0, 1) just like every other exponential, has horizontal asymptote y = 0, has range (0, ∞), and grows monotonically. Its graph sits between y = 2x and y = 3x, since 2 < e < 3.
The reason to single out e now, even before its calculus role makes sense, is that the inverse of ex — called the natural logarithm, written ln(x) — appears constantly on the CSCA exam. Any equation involving ex is most cleanly solved using ln, just as any equation involving 10x is most cleanly solved using log10. The next page introduces logarithms in detail.
Your turn! · Practice questions
Things to remember
| Concept | What to know |
|---|---|
| Definition | f(x) = ax with a > 0, a ≠ 1. The variable lives in the exponent. |
| Domain and range | Domain: all real numbers. Range: (0, ∞) — always positive, never zero. |
| y-intercept | Always (0, 1), since a0 = 1 for any base. |
| Growth (a > 1) | Increases as x increases. y → ∞ as x → ∞; y → 0 as x → −∞. |
| Decay (0 < a < 1) | Decreases as x increases. y → 0 as x → ∞; y → ∞ as x → −∞. |
| Asymptote | Basic curve has y = 0. Vertical shift moves it: y = ax + c has asymptote y = c. |
| Negative exponent | a−n = 1/an. Negative exponent makes a reciprocal, not a negative number. |
| Fractional exponent | am/n = (ⁿ√a)m. Take the n-th root, then raise to the m-th power. |
| Natural exponential | e ≈ 2.718. The function ex is exponential growth with base e; sometimes written exp(x). |
| Identity | (1/a)x = a−x. Decay with base 1/a is the same as growth with base a reflected across the y-axis. |