Express the function in the form f of g u(t) = sqrt(tan(t))

To express the function \( f(g(u(t))) = \sqrt{\tan(t)} \), we need to define functions \( f \), \( g \), and \( u \) such that:

\[
f(g(u(t))) = \sqrt{\tan(t)}
\]

We can do this by breaking down the function step by step:

1. **Innermost function \( u(t) \):** Start with the simplest expression, which is just \( t \) itself. So, let's set:
   \[
   u(t) = t
   \]

2. **Middle function \( g(u) \):** The next step is to apply the tangent function to \( u(t) \). Therefore, let:
   \[
   g(u) = \tan(u)
   \]

3. **Outermost function \( f(g) \):** Finally, we apply the square root to \( g(u) \). So, define:
   \[
   f(g) = \sqrt{g}
   \]

Now, combining these functions:

\[
f(g(u(t))) = f(\tan(u(t))) = f(\tan(t)) = \sqrt{\tan(t)}
\]

Thus, the functions are:

- \( u(t) = t \)
- \( g(u) = \tan(u) \)
- \( f(g) = \sqrt{g} \)

Would you like more details or have any other questions?

---

Here are 5 related questions:

1. How would you express \( f(g(u(t))) = \sin(\sqrt{t}) \) using similar steps?
2. What is the derivative of \( f(g(u(t))) = \sqrt{\tan(t)} \) with respect to \( t \)?
3. How would you express \( f(g(u(t))) = e^{\sin(t)} \) in terms of \( f \), \( g \), and \( u \)?
4. What happens if you change \( u(t) \) to \( u(t) = 2t \) in the original function?
5. How do you express \( f(g(u(t))) = \ln(\cos(t)) \)?

**Tip:** When breaking down a composite function, always start from the innermost operation and work your way outward to define each component function.

Learn how to express the composite function f(g(u(t))) = sqrt(tan(t)) step-by-step. Understand the concepts of composite functions, trigonometric functions, and square root functions in mathematics. Suitable for advanced high school students and anyone interested in mathematical functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation