Key Properties of Exponential Functions: Domain, Range, Intercepts and Asymptotes

A cup of hot chocolate

You come in from shovelling snow and make a mug of hot chocolate at 90°C. You get distracted and leave it on the counter. It cools quickly at first, then more and more slowly. Does it ever get colder than the room? No: it creeps toward room temperature but never goes below it.

That "creeping toward a line but never reaching it" is called an asymptote, and it is one of the key properties of every exponential function. In this lesson you will learn to describe exponential functions completely: domain, range, intercepts, increasing or decreasing, and asymptote.

The properties of y = aˣ

For every exponential function f(x) = aˣ with a > 0 and a ≠ 1:

• Domain: all real numbers, {x ∈ ℝ}. You can put any x in the exponent.
• Range: all positive real numbers, {y ∈ ℝ | y > 0}. A positive base raised to any power is positive.
• y-intercept: (0, 1), because a⁰ = 1.
• x-intercept: none, because aˣ is never 0.
• Increasing if a > 1; decreasing if 0 < a < 1.
• Horizontal asymptote: the x-axis, y = 0.

The notation {y ∈ ℝ | y > 0} is read "the set of all real numbers y such that y is greater than 0".

Why the curve never touches y = 0

On the left side of y = 2ˣ the values are 1/2, 1/4, 1/8, 1/16, … Each one is half of the one before. Halving a positive number always leaves a positive number, so the values get as close to 0 as you like but never reach it. That is exactly what a horizontal asymptote means.

For a decreasing function such as y = (1/2)ˣ, the same thing happens on the right side.

Exponential models: y = a(b)ˣ and y = a(b)ˣ + k

Real situations rarely start at exactly 1, so models usually look like y = a(b)ˣ:

• the initial value a is the y-intercept (when x = 0, y = a × b⁰ = a);
• the base b (the growth or decay factor) is the constant ratio in the table.

Some situations level off at a value other than 0, like a drink cooling to room temperature. Then the model looks like y = a(b)ˣ + k. Adding k moves the whole graph up k units, so the asymptote becomes y = k and the range becomes y > k (if a > 0).

The hot chocolate in the hook could be modelled by T = 20 + 70(0.85)ᵗ, where T is the temperature in °C and t is the time in minutes. (This is a model with made-up but realistic numbers.) At t = 0, T = 20 + 70 = 90°C. As t grows, 70(0.85)ᵗ shrinks toward 0, so T approaches 20°C, the room temperature.