the function f is given by f(θ)=tan(θ - π/4). which of the following descriptions is true for the graph of…

the function f is given by f(θ)=tan(θ - π/4). which of the following descriptions is true for the graph of f? a all asymptotes of the graph of f occur at input values θ = π/4+πk, where k is an integer. b all asymptotes of the graph of f occur at input values θ = 3π/4+πk, where k is an integer. c all asymptotes of the graph of f occur at input values θ = π/4+2πk, where k is an integer.

the function f is given by f(θ)=tan(θ - π/4). which of the following descriptions is true for the graph of f? a all asymptotes of the graph of f occur at input values θ = π/4+πk, where k is an integer. b all asymptotes of the graph of f occur at input values θ = 3π/4+πk, where k is an integer. c all asymptotes of the graph of f occur at input values θ = π/4+2πk, where k is an integer.

Answer

Explanation:

Step1: Recall tangent - asymptote property

The tangent function $y = \tan(x)$ has asymptotes at $x=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$.

Step2: Set the argument of the given tangent function equal to the asymptote formula

For the function $f(\theta)=\tan(\theta - \frac{\pi}{4})$, we set $\theta-\frac{\pi}{4}=\frac{\pi}{2}+k\pi$, where $k$ is an integer.

Step3: Solve for $\theta$

Add $\frac{\pi}{4}$ to both sides of the equation $\theta-\frac{\pi}{4}=\frac{\pi}{2}+k\pi$. $\theta=\frac{\pi}{2}+\frac{\pi}{4}+k\pi=\frac{2\pi + \pi}{4}+k\pi=\frac{3\pi}{4}+k\pi$.

Answer:

B. All asymptotes of the graph of $f$ occur at input values $\theta=\frac{3\pi}{4}+\pi k$, where $k$ is an integer.