Use the graph of the function to find the domain and range...

Use the graph of the function to find the domain and range of $f$ and the indicated function values.
$$\begin{array}{ll}{\text { (a) } f(-2)} & {\text { (b) } f(-1)} \\ {\text { (c) } f\left(\frac{1}{2}\right)} & {\text { (d) } f(1)}\end{array}$$

Key Concepts

Graph of a Function

A graph of a function is a visual representation of the relationship between inputs and outputs. It plots each input value along the x-axis and its corresponding output along the y-axis. By analyzing the graph, one can determine features such as the domain, range, intercepts, and specific function values at given inputs.

Domain

The domain of a function refers to the complete set of input values (typically x-values) for which the function is defined. When using a graph, the domain is determined by the horizontal spread of the graph, indicating the minimum and maximum x-values that the graph covers.

Range

The range of a function is the complete set of output values (f(x)-values) that the function can attain. From a graph, it is found by looking at the vertical extent – the minimum and maximum values that the function reaches.

Function Evaluation

Function evaluation involves substituting a specific input value into the function to obtain the corresponding output value. When provided with a graph, one identifies the point on the graph that aligns with the input value, then reads off the corresponding y-coordinate, which represents f(x) for that input.

Transcript

00:01 We're given the graph of a function, and we're asked to find the domain and range of the function f and the indicated function values.

00:13 The graph of the function is located following exercise 7 of this section.

00:24 So first to find the domain and range of our function, well, let's look at the x -axis of this graph.

00:36 We see that tracing the function, the function goes to both.

00:42 Positive and negative infinity along the x -axis.

00:48 So the domain of the function f is going to be an interval notation from negative infinity to infinity.

01:02 Of course, this can also be written as the set of all real numbers are.

01:07 Now to find the range of our function, let's look along the y -axis.

01:14 So we see that looking at the graph of our function, the function is always greater than or equal to y equals negative 4.

01:28 And it goes to positive infinity.

01:31 So our range of our function, this is going to be from negative 4 inclusive to positive infinity.

01:45 Notice that i've added the bracket here.

01:52 Now in part a, the function value is.

01:57 Is f of negative 2.

02:03 To find this value, we'll look at the graph of our function, find x equals negative 2 on the x -axis, and we'll trace up or down until we find where this intersects our graph.

02:17 We see that it intersects the graph at y equals 0.

02:21 And therefore, the value of our function, f at negative 2, is 0.

02:34 Next in part b, we're asked to find the value f of negative 1.

02:45 Excuse me.

02:48 So we'll do the same thing...

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