Notes on Functions and Graphs from Precalculus

Objectives of Functions and Graphs

  • Identify intervals on which a function increases, decreases, or is constant.

  • Use graphs to locate relative maxima or minima.

  • Test for symmetry to determine even or odd functions.

  • Understand and use piecewise functions.

  • Find and simplify a function’s difference quotient.

Increasing, Decreasing, and Constant Functions

  • Increasing Function: A function is increasing on an interval ( I ) if for any ( x1, x2 \in I ) where ( x1 < x2 ), then ( f(x1) < f(x2) ).

  • Decreasing Function: A function is decreasing on an interval ( I ) if for any ( x1, x2 \in I ) where ( x1 < x2 ), then ( f(x1) > f(x2) ).

  • Constant Function: A function is constant on an interval ( I ) if for any ( x1, x2 \in I ), then ( f(x1) = f(x2) ).

Relative Maximum and Minimum

  • Relative Maximum: A function value ( f(a) ) is a relative maximum if there exists an interval containing ( a ) such that ( f(a) > f(x) ) for all ( x \neq a ) in that interval.

  • Relative Minimum: A function value ( f(b) ) is a relative minimum if there exists an interval containing ( b ) such that ( f(b) < f(x) ) for all ( x \neq b ) in that interval.

Symmetry in Functions

  • Even Function: A function ( f ) is even if ( f(-x) = f(x) ) for all ( x ) in the domain. The graph is symmetric with respect to the y-axis.

  • Odd Function: A function ( f ) is odd if ( f(-x) = -f(x) ) for all ( x ) in the domain. The graph is symmetric with respect to the origin.

Piecewise Functions

  • A piecewise function is defined by two or more equations over specific domains. This allows different rules for different parts of the function.

Difference Quotient

  • The difference quotient for a function (f) is a way to measure how the function changes as we make small changes in its input. It is calculated using the formula:

  • [ \text{Difference Quotient} = \frac{f(x+h) - f(x)}{h} \text{ for } h \neq 0 ]

  • This means you take the value of the function at a point (x + h) and subtract the value at (x). Then, you divide by the small change (h) you made in the input.

  • As h gets smaller, this expression helps us understand how the function behaves and is used to find the derivative, which tells us the rate of change of the function at that point.