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. $f(-3)$ d. the values of $x$ for which $f(x)=-2$ 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 $f(-8)$ positive or negative?

Key Concepts

Inequalities Involving Functions

This concept involves determining the set of input values for which the function satisfies a particular inequality, such as being less than, greater than, or equal to a specific value. Analyzing these inequalities helps in identifying intervals on the graph where the function meets certain conditions.

Sign Analysis

Sign analysis is the process of determining whether the function outputs positive or negative values over different parts of its domain. By examining the intervals where the function is above or below the x-axis, one can understand and predict the behavior of the function, such as when evaluating the sign of the function at a specific input.

Intercepts

Intercepts are the points where the graph of the function intersects the coordinate axes. The x-intercepts indicate the zeros of the function, while the y-intercept is found by evaluating the function at x=0. Recognizing these points helps in graphically understanding the function's behavior.

Function Evaluation

Function evaluation involves substituting a specific value from the domain into the function to calculate its output value (for example, f(a)). This concept is essential not only for finding specific points on the graph but also for understanding how changes in the input affect the output.

Range

The range of a function is the complete set of all its possible output values (typically y-values). It reflects the vertical extent of the function's graph, indicating which values the function can produce. Representing the range using interval notation provides a clear view of the limits of the function's outputs.

Domain

The domain of a function refers to all the possible input values (typically x-values) for which the function is defined. Understanding the domain is fundamental when analyzing functions, as it tells you over which interval or set of values the function can be evaluated, and is often expressed using interval or set-builder notation.

Zeros of a Function

Zeros of a function are the x-values for which the function's output is zero. These points correspond to the x-intercepts on the graph where the function crosses the x-axis. Identifying these zeros is vital for solving equations and sketching the behavior of the function.

Transcript

00:01 All right, we're going to use the graph of this function in order to answer several questions.

00:07 First is the domain.

00:09 Those are your x values.

00:10 Notice, because of the arrowhead here, that it's going to go forever to the left.

00:15 And this arrowhead is going both up and at the right.

00:18 So because we have arrowheads going forever to the left and forever to the right with no gaps in between, we say that the domain is going to be all reels, from negative infinity to positive infinity.

00:32 The range are your y values.

00:35 So think smallest, lowest to the highest, from negative infinity to positive infinity.

00:40 The lowest that this graph goes is at negative 4.

00:48 So that is right here.

00:50 That's the smallest y value all the way up to positive infinity because this arrowhead is going up.

00:55 So we would say from negative 4 and it does include it does appear to hit that point.

01:00 So we say from negative 4 included to positive infinity.

01:10 F of negative 3, we want to find the graph's output when the input is negative 3.

01:17 So we find negative 3 on the x -axis, and we follow it up to the function, which has a y -value, an output of 4.

01:35 We want to know for what x values we have an function output of negative 2.

01:42 So it appears to happen in two locations.

01:46 We have an output of negative 2, add an input of positive 2, and positive 6.

01:56 Positive 2 and positive 6.

02:02 Because f of 2 is negative 2 and f of 6 is negative 2.

02:11 The points where the graph crosses the x -axis.

02:16 So here's our x -axis...

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