Math Problem Statement

Consider the table of values below for the function f(x)f(x)​. x12345f(x)111213141xf(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)​. 

Solution

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

\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.

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Math Problem Analysis

Mathematical Concepts

Functions
Linear Functions
Quadratic Functions

Formulas

Linear function form: f(x) = a + bx
Quadratic function form: xf(x) = ax + bx^2

Theorems

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Suitable Grade Level

Grades 7-9