College Algebra Module 2 Exam Study Guide – Step-by-Step Guidance

Q7. For the following graph of $f(x)$, find:

• (a) Domain

• (b) Range

• (c) Interval(s) where increasing

• (d) $f(0)$

• (e) The value(s) of $x$ for which $f(x) = -1$

Background

Topic: Graphs of Functions, Domain and Range, Function Evaluation, and Increasing/Decreasing Intervals

This question tests your ability to interpret a graph of a function and extract important information such as domain, range, intervals of increase, and specific function values.

Key Terms and Concepts:

• Domain: The set of all possible $x$-values for which the function is defined.

• Range: The set of all possible $y$-values (outputs) the function can take.

• Increasing Interval: Where the function values rise as $x$ increases.

• Function Evaluation: Finding $f(a)$ means finding the $y$-value when $x = a$.

• Solving $f(x) = k$: Find all $x$-values where the graph crosses the horizontal line $y = k$.

Step-by-Step Guidance

1. Domain: Look at the $x$-axis and determine the leftmost and rightmost $x$-values where the graph exists. Pay attention to whether endpoints are included (solid dot) or excluded (open dot).

2. Range: Examine the $y$-axis to find the lowest and highest $y$-values the graph attains. Again, check for inclusion/exclusion at endpoints.

3. Increasing Interval(s): Identify the interval(s) of $x$ where the graph rises as you move from left to right. Mark the $x$-values where this occurs.

4. Function Value at $x = 0$: Find the point on the graph where $x = 0$ and read the corresponding $y$-value.

5. Solving $f(x) = -1$: Draw or imagine a horizontal line at $y = -1$ and find all $x$-values where the graph intersects this line.

Try solving on your own before revealing the answer!

Final Answers:

• (a) Domain: $[-4, \infty)$

• (b) Range: $[-1, \infty)$

• (c) Increasing Interval(s): $(-2, \infty)$

• (d) $f(0)$: $1$

• (e) $f(x) = -1$ when $x = -2$

These answers are based on reading the graph: the domain starts at $x = -4$ (solid dot) and continues right, the lowest $y$-value is $-1$, the function increases after $x = -2$, $f(0)$ is the $y$-value at $x = 0$, and $f(x) = -1$ at $x = -2$.