exponential function
Imagine money in a bank that grows by the same percentage every year, or a rumor that spreads by doubling each day. The number does not climb by adding a fixed amount each step — it climbs by multiplying by a fixed factor each step. A function that captures this kind of repeated multiplication is an exponential function.
Formally, an exponential function has the form f(x) = b^x, where the base b is a positive constant not equal to 1, and the variable x sits up in the exponent. This is the key contrast with a power function like x^2, where the variable is the base and the exponent is fixed. As x increases by 1, the output is multiplied by b; if b > 1 the graph rises ever more steeply, and if 0 < b < 1 it falls toward zero.
Every exponential function f(x) = b^x passes through the point (0, 1), because b^0 = 1 for any allowed base. Its outputs are always positive, so the graph never touches the x-axis — the x-axis is a horizontal asymptote it approaches but never reaches. We exclude b = 1 (which would give the constant 1) and b ≤ 0 (which breaks down for fractional exponents).
For f(x) = 2^x: f(0) = 1, f(1) = 2, f(2) = 4, f(3) = 8. Each step multiplies the previous output by 2. Even f(-1) = 1/2 stays positive.
Equal steps in x produce equal multiplicative jumps in the output.
Do not confuse f(x) = 2^x (variable in the exponent, exponential) with f(x) = x^2 (variable in the base, a power function). They behave very differently: 2^x eventually outgrows x^2 no matter how big x^2 looks at first.