Exponential Functions
Exponential functions are a class of functions characterized by their rapid growth or decay. The general form of an exponential function is:
$f(x) = a^x$
where $a$ is a positive constant (a > 0) and $a \neq 1$. The base $a$ determines the behavior of the function:
- If $a > 1$, the function grows exponentially
- If $0< a < 1$, the function decays exponentially
For instance, $f(x) = 2^x$ is a growing exponential function, while $g(x) = (0.5)^x$ is a decaying exponential function.
Properties of of the Parent Exponential Function
- Domain: All real numbers
- Range: All positive real numbers (y > 0)
- y-intercept: Always (0, 1)
- Horizontal asymptote: y = 0
- Always positive
- One-to-one function (each x-value corresponds to a unique y-value)
The natural exponential function, $f(x) = e^x$, is a special case where the base is the mathematical constant e (≈ 2.71828). This function is particularly important in calculus and many real-world applications.
Graphing Exponential Functions
TipWhen graphing exponential functions, remember:
- For $a > 1$, the graph rises steeply to the right
- For $0< a < 1$, the graph falls steeply to the right, approaching but never touching the x-axis
To graph exponential functions that have transformations applied onto them, use the general form:
$$y = (a^{x-h})+ k$$
Where $h$ is the horizontal translation and $k$ is the vertical translation, with an asymptote at $y=k$.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. The general form of a logarithmic function is:
$f(x) = \log_a x$
where $a$ is the base of the logarithm and $a,x > 0$. The most common bases are:
- Base 10: $\log_{10} x$ (common logarithm, often written simply as $\log x$)
- Base e: $\ln x$ (natural logarithm)
Properties of Parent Logarithmic Function
- Domain: All positive real numbers (x > 0)
- Range: All real numbers
- x-intercept: (1, 0)
- Vertical asymptote: x = 0
