Which element of domain has the image 1 in the
function ?

Ask by Herrera Bird. in Nepal

Jan 02,2025

Question

Which element of domain has the image 1 in the function $$ f(x)=\frac{2 	an x}{1+	an x} $$ ?

UpStudy AI · Accepted Answer

To determine the values of $$ x $$ in the domain of the function $$ f(x) = \frac{2 \tan x}{1 + \tan x} $$ that map to the image $$ 1 $$, we need to solve the equation:

$$
\frac{2 \tan x}{1 + \tan x} = 1
$$

**Step 1: Solve for $$ \tan x $$**
$$
2 \tan x = 1 + \tan x
$$
$$
2 \tan x - \tan x = 1
$$
$$
\tan x = 1
$$

**Step 2: Find $$ x $$ such that $$ \tan x = 1 $$**
The general solution for $$ \tan x = 1 $$ is:
$$
x = \frac{\pi}{4} + k\pi \quad \text{where } k \text{ is any integer}
$$

**Step 3: Determine the Domain of $$ f(x) $$**
The function $$ f(x) $$ is defined as long as the denominator $$ 1 + \tan x $$ is not zero and $$ \tan x $$ itself is defined. Therefore, the domain excludes:
$$
x \neq \frac{\pi}{2} + k\pi \quad \text{and} \quad x \neq -\frac{\pi}{4} + k\pi \quad \text{for any integer } k
$$

**Conclusion:**
The elements of the domain that map to the image $$ 1 $$ under the function $$ f(x) $$ are all real numbers of the form:
$$
x = \frac{\pi}{4} + k\pi \quad \text{where } k \text{ is any integer}
$$

**Final Answer:**
All real numbers congruent to π⁄4 modulo π. In other words, the domain elements x satisfy
x = π/4 + k π for some integer k.
All real numbers of the form $$ x = \frac{\pi}{4} + k\pi $$ where $$ k $$ is any integer.

Which element of domain has the image 1 in the
function ?

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Which element of domain has the image 1 in the function ?