Obtaining Information from the Graph of a Function

The graph of \(y= f( x) \) is given in Figure 12. (\(x\) might represent time and \(y\) might represent the distance of the bob of a pendulum from its at-rest position. Negative values of \(y\) would indicate that the bob is to the left of its at-rest position; positive values of \(y\) would mean that the bob is to the right of its at-rest position.)

1. What are \(f( 0) \), \(f\left( \dfrac{3\pi }{2}\right) \), and \(f( 3\pi ) \)?
2. What is the domain of \(f\)?
3. What is the range of \(f\)?
4. List the intercepts of the graph.
5. How many times does the line \(y=2\) intersect the graph of \(f\)?
6. For what values of \(x\) does \(f(x) =-4\)?
7. For what values of \(x\) is \(f(x)>0\)?

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Solution (a) Since the point \( ( 0,4 ) \) is on the graph of \(f\), the \(y\)-coordinate \(4\) is the value of \(f\) at \(0;\) that is, \( f ( 0 ) =4\). Similarly, when \(x=\dfrac{3\pi }{2}\), then \(y=0\), so \( f \left( \dfrac{3\pi }{2} \right) =0\), and when \(x=3\pi \), then \(y=-4\), so \( f ( 3\pi) =-4\).

(b) The points on the graph of \(f\) have \(x\)-coordinates between \( 0 \) and \(4\pi \) inclusive. The domain of \(f\) is \( \{ x|0\leq x\leq 4\pi \} \) or the closed interval \( [ 0,4\pi ] \).

(c) Every point on the graph of \(f\) has a \(y\)-coordinate between \( -4\) and \(4\) inclusive. The range of \(f\) is \( \{ y|{-}4\leq y\leq 4 \} \) or the closed interval \( [ -4,4 ] .\)

(d) The intercepts of the graph of \(f\) are \( ( 0,4 ) \), \( \left( \dfrac{\pi }{2},0 \right) \), \( \left( \dfrac{3\pi }{2},0 \right) \), \( \left( \dfrac{5\pi }{2},0 \right) \), and \(\left( \dfrac{7\pi }{2},0\! \right) \).

(e) Draw the graph of the line \(y=2\) on the same set of coordinate axes as the graph of \(f\). The line intersects the graph of \(f\) four times.

(f) Find points on the graph of \(f\) for which \(y=f ( x ) =-4\); there are two such points: \( ( \pi ,-4 ) \) and \( ( 3\pi ,-4 ) .\) So \(f ( x ) =-4\) when \(x=\pi \) and when \( x=3\pi\).

(g) \(f ( x ) >0\) when the \(y\)-coordinate of a point \( ( x,y ) \) on the graph of \(f\) is positive. This occurs when \(x\) is in the set \( \left[ 0,\dfrac{\pi }{2} \right) \cup \left( \dfrac{3\pi }{2} ,\dfrac{5\pi }{2} \right) \cup \left( \dfrac{7\pi }{2},4\pi \right]\).