1.5 | Analyzing Graphs of Functions


  • A function \(f\) is increasing on an interval if, for any \(x_1\) and \(x_2\) in the interval, \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\).
  • A function \(f\) is decreasing on an interval if, for any \(x_1\) and \(x_2\) in the interval, \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\).
  • A function \(f\) is constant on an interval if, for any \(x_1\) and \(x_2\) in the interval, \(f(x_1) = f(x_2)\).
A function value \(f(a)\) is called a relative maximum of \(f\) if there exists an interval \((x_1, x_2)\) that contains \(a\) such that
$$
x_1 < x < x_2 \text{ implies } f(a) \leq f(x). $$ A function value \(f(a)\) is called a relative maximum of \(f\) if there exists an interval \((x_1, x_2)\) that contains \(a\) such that
$$
x_1 < x < x_2 \text{ implies } f(a) \geq f(x). $$
The average rate of change between any two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) is the slope of the line through the two points and is usually denoted \(f_{avg}\). The line through the two points is called a secant line, and the slope of the line is denoted \(m_{sec}\) and can be computed using the slope formula. Therefore,
$$
f_{avg} = m_{sec} = \frac{f(x_2) -f(x_1)}{x_2-x_1}
$$
A function \(y = f(x)\) is even if, for each \(x\) in the domain of \(f\),
$$
f(-x) = f(x).
$$
A function \(y = f(x)\) is odd if, for each \(x\) in the domain of \(f\),
$$
f(-x) = – f(x).
$$

E 1.5 Exercises