Question

The graph of $y=f\left(s\right)$ is shown above. If $-3\le x\le 6$, for how many values of $x$ does $f\left(x\right)=2$?

• A. None
• B. One
• C. Two
• D. Three
• E. More than three

Answer 1

The correct option is D Three

We know that

$f\left(x\right)$ $+$ $a$ translates $f\left(x\right)$ vertically

If the original function is $y$ $=$ $f\left(x\right)$, the translation of the function vertically upward or downward is the function $f\left(x\right)$ $+$ $a$.

If $a$ $>$ $0$, the graph translates upward by ${}^{\prime }{a}^{\prime }$ units.

If $a$ $<$ $0$, the graph translates downward by ${}^{\prime }{a}^{\prime }$ units.

Here,

$f\left(x\right)$ is the given in the figure and $a$ $=$ $-2$, $(a$ $<$ $0)$

As $(a$ $<$ $0)$, the graph translates downward.

By sliding the given graph downward by ${}^{\prime }{2}^{\prime }$ units, we will get $f\left(x\right)$ $-$ $2$ $=$ $0$.

After sliding the graph, we can see that the graph cuts the $x$-axis at $three$ points.

These $three$ points lie in the interval $-3$ $\le$ $x$ $\le$ $6$.

Therefore, $f\left(x\right)$ $=$ $2$ have three values of ${}^{\prime }{x}^{\prime }$ in the given interval.