Analyzing Features of a Piecewise Function Graph

Q1. Analyze the features of the given function graph:

• Identify all local maxima and minima.

• Find all x-intercepts and y-intercepts.

• State the domain and range using interval notation.

• List the intervals where $f(x) > 0$ and $f(x) < 0$.

• List the intervals where $f(x)$ is increasing and where it is decreasing.

Background

Topic: Features of Functions (Graph Analysis)

This question tests your ability to interpret a function's graph and identify key features such as intercepts, extrema (maxima and minima), domain, range, and intervals of positivity/negativity and monotonicity (increasing/decreasing behavior).

Key Terms and Formulas

• Local Maximum: A point where the function changes from increasing to decreasing, and is higher than nearby points.

• Local Minimum: A point where the function changes from decreasing to increasing, and is lower than nearby points.

• x-intercept: Where the graph crosses the x-axis ($f(x) = 0$).

• y-intercept: Where the graph crosses the y-axis ($x = 0$).

• Domain: All possible $x$-values for which the function is defined.

• Range: All possible $y$-values the function attains.

• Intervals of Increase/Decrease: Where the function is going up or down as $x$ increases.

• Intervals where $f(x) > 0$ or $f(x) < 0$: Where the graph is above or below the x-axis.

Step-by-Step Guidance

1. Examine the graph to identify any local maxima or minima. Look for points where the graph changes direction (from increasing to decreasing or vice versa).

2. Find the x-intercept(s) by locating where the graph crosses the x-axis. Write the x-value(s) where $f(x) = 0$.

3. Find the y-intercept by identifying where the graph crosses the y-axis ($x = 0$). Record the corresponding $y$ value.

4. Determine the domain by observing the leftmost and rightmost points on the graph. State the interval of $x$-values for which the graph exists.

5. Determine the range by finding the lowest and highest $y$-values the graph attains. State the interval of $y$-values covered by the graph.

6. Identify the intervals where $f(x) > 0$ (graph above the x-axis) and $f(x) < 0$ (graph below the x-axis). Use interval notation.

7. Analyze where the function is increasing (slopes upward as $x$ increases) and where it is decreasing (slopes downward as $x$ increases). State these intervals in terms of $x$.

Try solving on your own before revealing the answer!

Final Answer:

• Local maxima: none

• Local minima: none

• x-intercept(s): $x = -1$

• y-intercept: $y = 1$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, \infty)$

• Interval(s) where $f(x) > 0$: $(-1, \infty)$

• Interval(s) where $f(x) < 0$: $(-\infty, -1)$

• Increasing: $(-\infty, \infty)$

• Decreasing: none

The graph is always increasing, has no local maxima or minima, and crosses the x-axis at $x = -1$ and the y-axis at $y = 1$. The function is positive for $x > -1$ and negative for $x < -1$.