Question

the function $f$ is defined by the given equation. for which of the following values of $k$ does $f(k)=3k$?
a $\frac{59}{5}$
b $\frac{59}{2}$
c $\frac{177}{5}$
d $59$
$f(x)=|59 - 2x|$

Explanation:

Step1: Set up the equation

Since $f(k)=|59 - 2k|$ and $f(k)=3k$, we have $|59 - 2k|=3k$.

Step2: Consider two cases

Case 1: When $59 - 2k\geq0$ (i.e., $k\leq\frac{59}{2}$), the equation becomes $59 - 2k = 3k$.
Solve for $k$:
$59=3k + 2k$
$59 = 5k$
$k=\frac{59}{5}$
Check if it satisfies the condition $k\leq\frac{59}{2}$. Since $\frac{59}{5}=11.8$ and $\frac{59}{2}=29.5$, $\frac{59}{5}$ satisfies the condition.
Case 2: When $59 - 2k<0$ (i.e., $k>\frac{59}{2}$), the equation becomes $-(59 - 2k)=3k$.
Expand: $- 59+2k = 3k$.
Solve for $k$:
$3k-2k=-59$
$k=-59$. But $-59$ does not satisfy the condition $k>\frac{59}{2}$, so we discard this solution.

Answer:

A. $\frac{59}{5}$