Discover the key difference between exponential functions and polynomials. Learn the definition, identify what makes a function exponential, and understand the restrictions on the base.
Lesson 3.1 of Exponents & Logarithms in Math 30-1: Alberta curriculum lessons.
You've already worked with exponents before, things like 2^3 = 8 or 5^2 = 25. In those cases, the exponent was a fixed number. What makes exponential functions different and exciting is that the exponent is now the variable.
In plain English. Instead of asking "what is 2 to the power of 3?", we're now asking "what is 2 to the power of x?", where x can be anything.
Exponential Function. An exponential function is a continuous function of the form:
f(x) = c^x
Where.
What does "continuous" mean?. The graph has no breaks, holes, or jumps in it, it flows smoothly from left to right without lifting your pencil.
You might wonder why we use the letter c for the base instead of b. Later in this unit, we'll use the parameters a, b, h, and k to describe transformations of functions. Since b already has a job in that notation, we use c here to avoid confusion.
Common Mistake Alert!. Just because a function involves exponents doesn't make it exponential. The one thing you must check is: is the variable in the exponent position?
f(x) = 3^x
x is the exponent, 3 is the fixed base
f(x) = ((1)/(2))^x
x is the exponent, (1)/(2) is the fixed base
f(x) = 7^x
x is the exponent, 7 is the fixed base
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