Use the graph of f to determine each of the following. Where...

Use the graph of $f$ to determine each of the following. Where applicable, use interval notation. (GRAPH CAN'T COPY). a. the domain of $f$ b. the range of $f$ c. $f(-4)$ d. the values of $x$ for which $f(x)=-3$ e. the points where the graph of $f$ crosses the $x$ -axis f. the point where the graph of $f$ crosses the $y$ -axis g. values of $x$ for which $f(x)>0$ h. Is $f(-2)$ positive or negative?

Key Concepts

Sign Analysis

Sign analysis involves determining whether the function's output is positive or negative over certain intervals. By inspecting the graph, one can identify regions where the function is above the x-axis (positive values) or below it (negative values), which is critical for understanding the function's overall behavior.

Interval Notation

Interval notation is a concise way to represent subsets of the real number line, particularly useful when expressing the domain and range. It uses endpoints and interval symbols like parentheses and brackets to indicate whether endpoints are included in the set.

Y-Intercept

The y-intercept is the point where the graph of the function crosses the y-axis. This occurs when the input value is zero, and the corresponding output is found directly from the graph, providing insight into the function’s initial value.

X-Intercepts

X-intercepts, or roots of the function, are the points where the graph crosses or touches the x-axis. These are found by determining the input values for which the output value is zero, indicating where the function equals zero.

Graphical Equation Solving

Graphical equation solving is the process of finding the input values that yield a given output by using the graph. This method visually identifies where the function’s curve meets a horizontal line corresponding to the specified output value.

Function Evaluation

Function evaluation involves determining the output value of the function for a specific input. This is done by locating the point on the graph that corresponds to a particular x-value and reading off or computing the associated y-value.

Range

The range of a function is the set of output values, or y-values, that the function can produce. In graph analysis, it is obtained by observing the lowest and highest points and all the intermediate values the function achieves, which helps in understanding the behavior of the function.

Domain

The domain of a function is the complete set of input values, typically represented by the x-values, for which the function is defined. It is fundamental in understanding the limitations of where a function can operate and is often determined by looking at the graph to see the leftmost and rightmost points that the curve reaches.

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