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.)

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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 $\left[ 0,\dfrac{\pi }{2} \right) \cup \left( \dfrac{3\pi }{2} ,\dfrac{5\pi }{2} \right) \cup \left( \dfrac{7\pi }{2},4\pi \right]$.