Question

1. the graph of $f(x)=\tan(bx)$, where $b$ is a constant, is shown in the $xy$-plane. what is the value of $b$?

(a) 4
(b) 2
(c) $\frac{\pi}{2}$
(d) $\frac{\pi}{4}$

1. the figures shows the graph of $f(x)=a\tan(bx)$, where $a$ and $b$ are constants, in the $xy$-plane. the graph of $f$ has two vertical asymptotes at $x=-\pi$ and $x=\pi$, and a point with coordinates given is on the graph of $f$. what are all solutions to $f(x)=1.829$

(a) $x=2.317$ only
(b) $x=2.317+\pi k$, where $k$ is any integer
(c) $x=2.317+2\pi k$, where $k$ is any integer
(d) $x=2.317+3\pi k$, where $k$ is any integer

Explanation:

Question 16

Step1: Find period of $f(x)=\tan(bx)$

From the graph, the distance between two consecutive vertical asymptotes is $2$. For $\tan(bx)$, period $T=\frac{\pi}{|b|}$. Here $T=2$.

Step2: Solve for $b$

Set $\frac{\pi}{b}=2$ (since $b>0$ for standard tangent graph direction). Rearrange to get $b=\frac{\pi}{2}$.

Step1: Find period of $f(x)=a\tan(bx)$

Vertical asymptotes at $x=-\pi$ and $x=\pi$, so period $T=\pi - (-\pi)=2\pi$. For tangent functions, solutions repeat every full period.

Step2: Identify solution pattern

Given one solution is $x=2.317$, all solutions are this value plus integer multiples of the period $2\pi$.

Answer:

$\boldsymbol{\frac{\pi}{2}}$ (Option C)

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Question 17