Use the graph of the function to answer (a)-(e). (a) Find...

Use the graph of the function to answer (a)-(e). (a) Find the domain and range of $f$. (b) Find the zero(s) of $f$. (c) Determine the intervals over which $f$ is increasing, decreasing, or constant. (d) Approximate any relative minimum or relative maximum values of $f$. (e) Is $f$ even, odd, or neither?

Use the graph of the function to answer (a)-(e).
(a) Find the domain and range of $f$.
(b) Find the zero(s) of $f$.
(c) Determine the intervals over which $f$ is increasing, decreasing, or constant.
(d) Approximate any relative minimum or relative maximum values of $f$.
(e) Is $f$ even, odd, or neither?

Key Concepts

Domain and Range

The domain of a function consists of all the possible input values (typically x-values) for which the function is defined, while the range is the set of all possible outputs (typically y-values) the function can produce. When analyzing a graph, the domain is observed by looking at the horizontal extent of the graph, and the range is determined by the vertical spread of the values that the graph attains.

Zeros or Roots

Zeros (or roots) of a function are the values of x at which the function equals zero, meaning the points where the graph crosses or touches the x-axis. Identifying the zeros is important for understanding the function's behavior, solving equations, and analyzing the function's intercepts with the axis.

Intervals of Increase and Decrease

Intervals where a function is increasing, decreasing, or constant are determined by examining the slope of the graph. A function is increasing if the graph moves upward as x increases, decreasing if it moves downward, and constant if it remains flat. These intervals help in understanding the overall trend and behavior of the function, and are essential in calculus for determining the derivative's sign.

Relative Extrema

Relative (or local) extrema refer to the points on the graph where the function attains a local minimum or maximum compared to the nearby points. These points are significant because they can indicate optimal values in various contexts. In a graph, relative minima and maxima are often identified at the turning points where the direction of the graph changes from increasing to decreasing or vice versa.

Even and Odd Functions

Even functions are those that exhibit symmetry about the y-axis, meaning f(x) = f(–x) for all x in the domain, while odd functions are symmetric about the origin, satisfying f(–x) = –f(x). Examining the symmetry of the graph helps determine whether the function is even, odd, or neither, and this symmetry has implications in simplifying function analysis and integration.

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