If a function is given by the formula \(y=f(x)\), then \(x\) is the independent variable and denotes the input; \(y\) is the dependent variable and denotes the output; the domain is the set of all possible inputs \(x\); the range is the set of all possible outputs \(y\).
The net change in the value of the function \(f\) between \(x=a\) and \(x=b\) is
The graph of a function \(f\) is the graph of the equation \(y=f(x)\) that defines \(f\).
A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the graph more than once.
A relation is any collection of ordered pairs \((x, y)\). The \(x\)-values are inputs and the corresponding \(y\)-values are outputs. A relation is a function if every input corresponds to exactly one output.
A function \(f\) is increasing on an interval if \(f\left(x_{1}\right)<f\left(x_{2}\right)\) whenever \(x_{1}<x_{2}\) in the interval. A function \(f\) is decreasing on an interval if \(f\left(x_{1}\right)>f\left(x_{2}\right)\) whenever \(x_{1}<x_{2}\) in the interval.
The function value \(f(a)\) is a local maximum value of the function \(f\) if \(f(a) \geq f(x)\) for all \(x\) near \(a\). In this case we also say that \(f\) has a local maximum at \(x=a\).
The function value \(f(b)\) is a local minimum value of the function \(f\) if \(f(b) \leq f(x)\) for all \(x\) near \(b\). In this case we also say that \(f\) has a local minimum at \(x=b\).
The average rate of change of the function \(f\) between \(x=a\) and \(x=b\) is the slope of the secant line between \((a, f(a))\) and \((b, f(b))\) :
A linear function is a function of the form \(f(x)=a x+b\). The graph of \(f\) is a line with slope \(a\) and \(y\)-intercept \(b\). The average rate of change of \(f\) has the constant value \(a\) between any two points.
Let \(c\) be a positive constant. To graph \(y=f(x)+c\), shift the graph of \(y=f(x)\) upward by \(c\) units.
To graph \(y=f(x)-c\), shift the graph of \(y=f(x)\) downward by \(c\) units.
To graph \(y=f(x-c)\), shift the graph of \(y=f(x)\) to the right by \(c\) units.
To graph \(y=f(x+c)\), shift the graph of \(y=f(x)\) to the left by \(c\) units.
To graph \(y=-f(x)\), reflect the graph of \(y=f(x)\) about the \(x\)-axis.
To graph \(y=f(-x)\), reflect the graph of \(y=f(x)\) about the \(y\)-axis.
If \(c>1\), then to graph \(y=c f(x)\), stretch the graph of \(y=f(x)\) vertically by a factor of \(c\).
If \(0<c<1\), then to graph \(y=c f(x)\), shrink the graph of \(y=f(x)\) vertically by a factor of \(c\).
If \(c>1\), then to graph \(y=f(c x)\), shrink the graph of \(y=f(x)\) horizontally by a factor of \(1 / c\).
If \(0<c<1\), then to graph \(y=f(c x)\), stretch the graph of \(y=f(x)\) horizontally by a factor of \(1 / c\).
A function \(f\) is
for every \(x\) in the domain of \(f\).
Given two functions \(f\) and \(g\), the composition of \(f\) and \(g\) is the function \(f \circ g\) defined by
The domain of \(f \circ g\) is the set of all \(x\) for which both \(g(x)\) and \(f(g(x))\) are defined.
A function \(f\) is one-to-one if \(f\left(x_{1}\right) \neq f\left(x_{2}\right)\) whenever \(x_{1}\) and \(x_{2}\) are different elements of the domain of \(f\).
A function is one-to-one if and only if no horizontal line intersects its graph more than once.
Let \(f\) be a one-to-one function with domain \(A\) and range \(B\). The inverse of \(f\) is the function \(f^{-1}\) defined by
The inverse function \(f^{-1}\) has domain \(B\) and range \(A\). The functions \(f\) and \(f^{-1}\) satisfy the following cancellation equations: