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Beginning Activity Beginning Activity 2: Functions and Intervals
A4US

Let g:R→R be defined by g(x)=x2, for each x∈R.

1.

We will first determine where g maps the closed interval [1,2]. (Recall that [1,2]={x∈R∣1≤x≤2}.) That is, we will describe, in simpler terms, the set {g(x)∣x∈[1,2]}. This is the set of all images of the real numbers in the closed interval [1,2].

(a)

Draw a graph of the function g using −3≤x≤3.

(b)

On the graph, draw the vertical lines x=1 and x=2 from the x-axis to the graph. Label the points P(1,f(1)) and Q(2,f(2)) on the graph.

(c)

Now draw horizontal lines from the points P and Q to the y-axis. Use this information from the graph to describe the set {g(x)∣x∈[1,2]} in simpler terms. Use interval notation or set builder notation.

2.

We will now determine all real numbers that g maps into the closed interval [1,4]. That is, we will describe the set {x∈R∣g(x)∈[1,4]} in simpler terms. This is the set of all preimages of the real numbers in the closed interval [1,4].

(a)

Draw a graph of the function g using −3≤x≤3.

(b)

On the graph, draw the horizontal lines y=1 and y=4 from the y-axis to the graph. Label all points where these two lines intersect the graph.

(c)

Now draw vertical lines from the points in Task 2.b to the x-axis, and then use the resulting information to describe the set {x∈R∣g(x)∈[1,4]} in simpler terms. (You will need to describe this set as a union of two intervals. Use interval notation or set builder notation.)