https://file.aiquickdraw.com/mathbot/file/1726458116372q6uufrw7.jpg

The problem in the image asks to find the range of the function:

\[
f(x) = 8 + 2 \cdot \arctan(7x + 3)
\]

Let's break this down step by step.

### Step 1: Understanding the function

The function involves an arctangent term:

\[
f(x) = 8 + 2 \arctan(7x + 3)
\]

The arctangent function, \(\arctan(x)\), has a well-known range:

\[
\arctan(x) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)
\]

### Step 2: Scaling the arctangent

Since \(\arctan(7x + 3)\) is just a shifted input to the arctangent function, the output range of \(\arctan(7x + 3)\) is still the same as for \(\arctan(x)\), that is:

\[
\arctan(7x + 3) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)
\]

Next, multiply the arctangent by 2:

\[
2 \cdot \arctan(7x + 3) \in \left( -\pi, \pi \right)
\]

### Step 3: Adding the constant

Now, add 8 to the entire expression:

\[
f(x) = 8 + 2 \cdot \arctan(7x + 3) \in \left( 8 - \pi, 8 + \pi \right)
\]

Thus, the range of \(f(x)\) is:

\[
(a, b) = \left( 8 - \pi, 8 + \pi \right)
\]

### Step 4: Approximating the values

Since \(\pi \approx 3.1416\):

\[
a = 8 - \pi \approx 8 - 3.1416 = 4.8584
\]
\[
b = 8 + \pi \approx 8 + 3.1416 = 11.1416
\]

### Final Answer:
The range of the function \(f(x)\) is approximately:

\[
(4.8584, 11.1416)
\]

Would you like a detailed explanation of any specific part? Here are 5 related questions to expand on the topic:

1. What is the general behavior of the \(\arctan(x)\) function?
2. How does scaling affect the range of a trigonometric function?
3. Why is the range of \(\arctan(x)\) limited to \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)?
4. How do you handle functions that involve more complex transformations of \(x\)?
5. How can you derive the inverse function of a trigonometric expression like this?

**Tip:** For any inverse trigonometric function like \(\arctan(x)\), always start by identifying the standard range before applying transformations like scaling and shifting.

Find the range of f(x) = 8 + 2 * arctan(7x + 3). The range is of the form (a, b). Enter the values of a and b.

Learn how to find the range of the function f(x) = 8 + 2 * arctan(7x + 3), a problem involving inverse trigonometric functions and transformations. Suitable for college-level calculus or advanced high school students.

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