Beginning Activity Beginning Activity 2: Functions and Intervals
Let \(g\x \mathbb{R} \to \mathbb{R}\) be defined by \(g ( x ) = x^2\text{,}\) for each \(x \in \mathbb{R}\text{.}\)
1.
We will first determine where \(g\) maps the closed interval \(\left[ 1, 2 \right]\text{.}\) (Recall that \([1, 2] = \left\{ x \in \R \mid 1 \leq x \leq 2 \right\}\text{.}\)) That is, we will describe, in simpler terms, the set \(\left\{ g ( x ) \mid x \in \left[ 1, 2 \right] \right\}\text{.}\) This is the set of all images of the real numbers in the closed interval \(\left[ 1, 2 \right]\text{.}\)
(a)
Draw a graph of the function \(g\) using \(-3 \leq x \leq 3\text{.}\)
(b)
On the graph, draw the vertical lines \(x = 1\) and \(x = 2\) from the \(x\)-axis to the graph. Label the points \(P \!\left(1, f ( 1 ) \right)\) and \(Q \!\left(2, f ( 2 ) \right)\) 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 \(\left\{ g ( x ) \mid x \in \left[ 1, 2 \right] \right\}\) 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 \(\left[ 1, 4 \right]\text{.}\) That is, we will describe the set \(\left\{ x \in \mathbb{R} \mid g ( x ) \in \left[ 1, 4 \right] \right\}\) in simpler terms. This is the set of all preimages of the real numbers in the closed interval \(\left[ 1, 4 \right]\text{.}\)
(a)
Draw a graph of the function \(g\) using \(-3 \leq x \leq 3\text{.}\)
(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 TaskĀ 2.b to the \(x\)-axis, and then use the resulting information to describe the set \(\left\{ x \in \mathbb{R} \mid g ( x ) \in \left[ 1, 4 \right] \right\}\) in simpler terms. (You will need to describe this set as a union of two intervals. Use interval notation or set builder notation.)