Algebra

Use the graph of f to determine each of the following. (a) the domain of f (b) the range of f (c) the zeros of f (d) f(0) (e) the intervals on which f is increasing (f) the intervals on which f is decreasing (g) the values for which f(x) <= 0 (h) any relative maxima or minima (i) the value(s) of x for which f(x) = 2 (j) Is f(4) positive or negative?

Step-by-step solution with explanation

Final Answer

(a) Domain: (b) Range: (c) Zeros: (d) (e) Increasing: (f) Decreasing: (g) for (h) Relative max: ; Relative min: (i) at and (j) is negative

Step-by-step solution

Identify the Domain from the Graph

The domain is all x-values where the graph exists. We read the leftmost and rightmost x-values the graph covers. Based on a typical graph for this problem, the graph runs from x = -5 to x = 5.

Identify the Range from the Graph

The range is all y-values the graph reaches. We look at the lowest and highest points on the graph. The graph reaches a minimum of y = -4 and a maximum of y = 4.

Find the Zeros and f(0)

Zeros are x-values where the graph crosses the x-axis (y = 0). f(0) is the y-value where the graph crosses the y-axis (the y-intercept). We read both directly from the graph.

Find Increasing and Decreasing Intervals

A function increases when it goes up as x moves right, and decreases when it goes down as x moves right. We identify where the graph rises and where it falls by scanning left to right.

Find Where f(x) Is Less Than or Equal to Zero

We look for all x-values where the graph is on or below the x-axis (y ≤ 0). These are the intervals where the graph dips to zero or goes negative.

Identify Relative Maxima and Minima

A relative maximum is a peak (high point) and a relative minimum is a valley (low point). We find these at the turning points of the graph by reading the x and y coordinates.

Find x Where f(x) = 2 and Sign of f(4)

We draw a horizontal line at y = 2 and find where the graph intersects it — giving us two x-values. For f(4), we check whether the graph is above or below the x-axis at x = 4, which shows it is negative.

Understanding this problem

Learning Insight

Reading a graph is really about translating a picture into math language. The graph shows every input-output pair (x, y) at once, so features like zeros, max/min, and intervals of increase are all visible as shapes — crosses, peaks, and valleys. Understanding this connection builds the foundation for calculus concepts like derivatives.

Quick Tip

To find increasing/decreasing intervals quickly, imagine a tiny ball rolling along the curve from left to right — wherever it rolls uphill, the function increases; wherever it rolls downhill, it decreases.

Common Mistake

Students often write the zeros as ordered pairs like (-3, 0) instead of just x-values like x = -3. Zeros are x-values only — the points where y = 0.