Consider the table of values below for the function f(x)*f*(*x*)​.
x12345f(x)111213141*xf*(*x*)​11​211​321​431​541​​

What type of function is f(x)*f*(*x*)​?


Linear

Exponential

Quadratic

​


Enter an explicit function for f(x)*f*(*x*)​. 

To determine the type of function \( f(x) \) from the given table of values, let's first analyze the pattern in the table:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & xf(x) \\
\hline
1 & 11 & 11 \\
2 & 12 & 24 \\
3 & 13 & 39 \\
4 & 14 & 56 \\
5 & 15 & 75 \\
\hline
\end{array}
\]

### Step 1: Identifying the Type of Function

We can see that \( xf(x) \) column values do not correspond to a straightforward pattern of addition (which would indicate a linear function), multiplication by a fixed factor (exponential function), or a pattern consistent with a quadratic function's form \( ax^2 + bx + c \). However, let's explore further to see the relationship between \( x \) and \( f(x) \).

### Step 2: Analyzing \( f(x) \)

From the table, \( f(x) \) increases linearly with \( x \):

- \( f(1) = 11 \)
- \( f(2) = 12 \)
- \( f(3) = 13 \)
- \( f(4) = 14 \)
- \( f(5) = 15 \)

It appears that \( f(x) \) can be described as \( f(x) = 10 + x \), which is a linear function.

### Step 3: Deriving \( xf(x) \)

Now, let's use this linear form to find an explicit function for \( xf(x) \).

Given \( f(x) = 10 + x \):

\[
xf(x) = x(10 + x) = 10x + x^2
\]

### Conclusion: Type of Function and Explicit Formula

1. **Type of Function**: Since \( xf(x) = 10x + x^2 \) is a polynomial function of degree 2 (quadratic), the type of function is **Quadratic**.

2. **Explicit Function**: The explicit function for \( xf(x) \) is:

\[
xf(x) = 10x + x^2
\]

Would you like more details on any step or have any questions?

Here are some related questions you might find interesting:

1. What are the properties of quadratic functions, and how do they differ from linear functions?
2. How would the graph of the function \( xf(x) = 10x + x^2 \) look like?
3. Can you find the vertex of the quadratic function \( xf(x) = 10x + x^2 \)?
4. How would you determine if a function is linear, quadratic, or exponential just by looking at a table of values?
5. What are some real-world applications of quadratic functions?

**Tip:** When examining tables of values to determine the type of function, look for consistent differences or ratios in values to identify patterns indicative of linear, quadratic, or exponential functions.

Learn how to identify the type of function f(x) from a table of values and derive its explicit form. This problem explores the concepts of linear and quadratic functions using a detailed step-by-step analysis.

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