https://r2.erweima.ai/mathbot/image/17258428373550q86w69u.jpg

The image presents a series of questions related to the function \( y = t^4 \). Let me guide you through solving each part:

### a) ¿Qué tipo de función es?
The function \( y = t^4 \) is a **polynomial function**. Specifically, it is a monomial with an even degree (degree 4), which makes it a type of **even function**. 

### b) Dominio
The **domain** of the function refers to the set of values that \( t \) can take. Since this is a polynomial function, it is defined for all real numbers. Hence, the domain is:
\[
\text{Dominio: } (-\infty, \infty)
\]

### c) Rango
The **range** is the set of possible values that \( y \) can take. Since \( t^4 \geq 0 \) for all \( t \), the range is:
\[
\text{Rango: } [0, \infty)
\]

### d) Si es creciente y/o decreciente y en qué intervalos
To determine where the function is increasing or decreasing, we need to find its derivative:
\[
y' = 4t^3
\]
- When \( t > 0 \), \( y' > 0 \), so the function is **increasing** in the interval \( (0, \infty) \).
- When \( t < 0 \), \( y' < 0 \), so the function is **decreasing** in the interval \( (-\infty, 0) \).

Thus, the function decreases in \( (-\infty, 0) \) and increases in \( (0, \infty) \).

### e) Si es continua o discontinua
A polynomial function is always continuous. Therefore, \( y = t^4 \) is **continuous** for all real numbers.

### f) Intersección y asíntotas
- **Intersección con el eje \( y \)**: At \( t = 0 \), \( y = 0^4 = 0 \), so the function intersects the \( y \)-axis at \( (0, 0) \).
- **Asymptotes**: Since the function approaches infinity as \( t \to \infty \) or \( t \to -\infty \), it does not have horizontal or vertical asymptotes.

### g) Realice una tabla de valores y grafique la función
A table of values can be created to plot the graph of the function. Here's a sample of values:

| \( t \) | \( y = t^4 \) |
|--------|--------------|
| -2     | 16           |
| -1     | 1            |
| 0      | 0            |
| 1      | 1            |
| 2      | 16           |

The graph of \( y = t^4 \) is symmetric with respect to the \( y \)-axis and has a parabolic shape that flattens near the origin and grows rapidly for large values of \( t \).

Would you like to see a graph of this function or go into more details about any part?

### Related Questions:
1. What is the behavior of polynomial functions of different even degrees?
2. How do you find the critical points of a polynomial function?
3. Why does the function \( t^4 \) have no vertical or horizontal asymptotes?
4. How can you verify the increasing and decreasing intervals using a second derivative?
5. How does the derivative of \( t^4 \) relate to the curvature of its graph?

#### Tip: 
When analyzing functions, always begin by identifying the type of function (e.g., polynomial, trigonometric, logarithmic), as it helps predict behavior like continuity, range, and intervals of increase/decrease.

Para la siguiente función: y = t^4. Determine y argumente con los procedimientos respectivos:
 a) ¿Qué tipo de función es?
 b) Dominio.
 c) Rango.
 d) Si es creciente y/o decreciente y en qué intervalos.
 e) Si es continua o discontinua.
 f) Intersección y asíntotas.
 g) Realice una tabla de valores (por lo menos 5), encuentre el vértice y grafique la función.

Learn how to analyze the polynomial function y = t^4, including finding its domain, range, intervals of increase and decrease, continuity, intercepts, asymptotes, and graphing it with a table of values. Suitable for high school math students in Grades 9-12.

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