An exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.
With base 2, every step right doubles y; with base 12, every step right halves it. Both start at (0, 1) because b0 = 1, and neither ever reaches 0.
Formal definition and notation
An exponential function is a function of the form f(x) = a · bx, where a is a nonzero real number (the initial value f(0)), b is a positive real number with b ≠ 1 (the base or growth factor) and x is any real number. Its domain is all real numbers. For a > 0 its range is y > 0 and its graph has the horizontal asymptotey = 0.
Symbol
Read it as
Meaning
f(x) = abx
f of x equals a times b to the x
general form; a is the starting value, b the factor per step
From two points one unit apart, b = f(x + 1)f(x). From a percent rate, b = 1 + r for growth and b = 1 − r for decay.
Try it: the input is the exponent
Interactive
Hold the base at 3 and step the exponent from −4 up to 6. The exponent is the input x of f(x) = 3x, and the value is the output: each step multiplies it by 3.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Exponential Function examples, step by step
Example 1: Evaluate an exponential function
Warm-up
For f(x) = 5 · 2x, find f(0), f(3) and f(−1).
f(0) = 5 · 20 = 5 · 1 = 5.
f(3) = 5 · 23 = 5 · 8 = 40.
f(−1) = 5 · 2−1 = 5 · 12 = 2.5.
Answer: f(0) = 5, f(3) = 40 and f(-1) = 2.5.
Example 2: Linear or exponential?
Core
A table shows x = 0, 1, 2, 3 and y = 6, 9, 13.5, 20.25. Is the relationship linear or exponential? Write its equation.
Differences: 9 − 6 = 3, 13.5 − 9 = 4.5. They are not equal, so it is not linear.
Ratios: 96 = 1.5, 13.59 = 1.5, 20.2513.5 = 1.5. They are equal, so it is exponential with b = 1.5.
The value at x = 0 is a = 6.
Equation: y = 6(1.5)x.
Answer: It is exponential: y = 6(1.5)^x.
Example 3: Equation from two points
Test-style
An exponential function y = abx passes through (0, 3) and (2, 48). What is y when x = 3?
At x = 0: y = a · b0 = a, so a = 3.
At x = 2: 3b2 = 48, so b2 = 16 and b = 4 (the base must be positive).
The function is y = 3 · 4x.
At x = 3: y = 3 · 64 = 192.
Answer: y = 192 when x = 3 (the function is y = 3 · 4x).
Common mistakes
✗ MistakeTreating y = 2x like y = x2: at x = 5, 25 = 10 or 25.
✓ Instead25 = 2 · 2 · 2 · 2 · 2 = 32.
In an exponential function the input is the exponent and the base is fixed. 25 means five factors of 2, not 2 times 5 or 5 squared.
✗ Mistake3 · 2x at x = 4: 64 = 1296.
✓ Instead3 · 24 = 3 · 16 = 48.
Exponents come before multiplication. The exponent belongs only to the base 2, not to the coefficient 3.
✗ MistakeA table that grows by 4, then 8, then 16 is linear because it always increases.
✓ InsteadIt is exponential: the ratios are constant (each difference doubles, so each y doubles).
Linear means equal differences; exponential means equal ratios. Increasing is not enough to decide.
Not to be confused with
Exponential Functionvs.Power function
In a power function such as y = x3 the base is the variable and the exponent is fixed. In an exponential function such as y = 3x the base is fixed and the exponent is the variable.
A linear function adds the same amount per step (y = 2x + 5: plus 2). An exponential function multiplies by the same factor per step (y = 5 · 2x: times 2).
A geometric sequence is an exponential function whose inputs are only the whole numbers 1, 2, 3, and so on. The exponential function also accepts fractions and negatives.
Quick check: 3 questions
Question 1 of 3
Which equation is an exponential function?
Show answer and explanation
Answer: C. Only y = 4(0.6)x has the variable in the exponent with a fixed positive base. y = x4 is a power function, y = 4x is linear and y = 4x2 + 1 is quadratic.
Question 2 of 3
What is the y-intercept of y = 7 · 3x?
Show answer and explanation
Answer: D. At x = 0, 30 = 1, so y = 7. 21 is the value at x = 1, 3 is the base, and 0 is the asymptote, which the graph never reaches.
Question 3 of 3
The table has x = 0, 1, 2 and y = 200, 100, 50. Which function fits?
Show answer and explanation
Answer: A. Each step multiplies y by 100200 = 0.5 and the start is 200. 200 − 50x gives 150 at x = 1; base 2 grows instead of halving; the last choice swaps a and b.
Exponential Function: frequently asked questions
How do you know if a function is exponential?
Look for the variable in the exponent, as in y = 5(1.2)x. In a table with equally spaced inputs, divide each output by the one before it. If the ratio is always the same, the function is exponential. If the differences are always the same instead, it is linear.
Why can the base of an exponential function not be negative or 1?
With base 1, 1x = 1 for every x, so the function is a constant, not exponential. With a negative base, values such as (−4)12 are not real numbers, so the graph would be full of gaps. Requiring b > 0 and b ≠ 1 keeps the function defined and always changing.
Does an exponential graph ever touch the x-axis?
No. For y = abx with a > 0, every output is positive, because a positive base to any power is positive. The graph gets closer and closer to y = 0 on one side, which makes the x-axis a horizontal asymptote, but it never reaches it. Adding a constant, as in y = 2x − 3, moves the asymptote to y = −3.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-LE.A.1, CCSS HSF-LE.A.2, CCSS HSF-IF.C.7.e
Cite this entry
Effortless Math Team. “Exponential Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/exponential-function/
Flashcard
TermExponential Functionf(x) = a · bx
DefinitionA function with the input in the exponent: start at a, multiply by b for each step. Example: y = 5 · 2x gives 5, 10, 20, 40.
FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2AsymptoteAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1x has both: x = 0 and y = 0.Functions · Algebra 2, Precalculus, CalculusEuler's NumberEuler’s number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1n )n gets closer and closer to as n grows. It is the base of natural exponential growth, ex, and of the natural logarithm, lnx.Functions · Algebra 2, Precalculus, CalculusPeriodThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.Trigonometry · Algebra 2, PrecalculusExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Exponential GrowthExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Functions · Algebra 1, Algebra 2Exponential DecayExponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.Functions · Algebra 1, Algebra 2PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2LogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusDifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · rn − 1. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, PrecalculusNegative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1Compound InterestCompound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + rn )nt, where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.Functions · Grade 7, Algebra 1, Algebra 2