Finish The Following Table For The Given Function With $x$ As The Independent Variable.$\[ \begin{tabular}{|l|l|} \hline $x$ & $f(x)$ \\ \hline 0 & 10 \\ \hline 5 & \\ \hline 10 & \\ \hline 15 & \\ \hline &

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Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function is often represented as a mathematical expression or a table. In this article, we will explore a given function table and complete it with the missing values.

The Function Table

xx f(x)f(x)
0 10
5
10
15

Understanding the Function

To complete the function table, we need to understand the relationship between the input values (x) and the output values (f(x)). However, the given function is not explicitly defined, so we will assume a linear function for simplicity.

Assuming a Linear Function

A linear function can be represented as:

f(x)=mx+bf(x) = mx + b

where m is the slope and b is the y-intercept.

Finding the Slope (m)

To find the slope, we need at least two points on the function. We can use the given points (0, 10) and (5, ?) to find the slope.

Let's assume the point (5, y) is on the function. We can use the slope formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1, y1) = (0, 10) and (x2, y2) = (5, y).

Substituting the values, we get:

m=y−105−0m = \frac{y - 10}{5 - 0}

Finding the Y-Intercept (b)

The y-intercept (b) is the value of f(x) when x = 0. We are given that f(0) = 10, so the y-intercept is:

b=10b = 10

Completing the Function Table

Now that we have the slope (m) and the y-intercept (b), we can complete the function table.

Using the linear function:

f(x)=mx+bf(x) = mx + b

Substituting the values of m and b, we get:

f(x)=y−105x+10f(x) = \frac{y - 10}{5}x + 10

We can now find the values of f(x) for x = 5, 10, and 15.

Calculating f(5)

Substituting x = 5 into the function, we get:

f(5)=y−105⋅5+10f(5) = \frac{y - 10}{5} \cdot 5 + 10

Simplifying the equation, we get:

f(5)=y−10+10f(5) = y - 10 + 10

f(5)=yf(5) = y

Since we don't know the value of y, we will leave it as a variable for now.

Calculating f(10)

Substituting x = 10 into the function, we get:

f(10)=y−105⋅10+10f(10) = \frac{y - 10}{5} \cdot 10 + 10

Simplifying the equation, we get:

f(10)=2y−20+10f(10) = 2y - 20 + 10

f(10)=2y−10f(10) = 2y - 10

Calculating f(15)

Substituting x = 15 into the function, we get:

f(15)=y−105⋅15+10f(15) = \frac{y - 10}{5} \cdot 15 + 10

Simplifying the equation, we get:

f(15)=3y−30+10f(15) = 3y - 30 + 10

f(15)=3y−20f(15) = 3y - 20

The Completed Function Table

xx f(x)f(x)
0 10
5 y
10 2y - 10
15 3y - 20

Discussion

In this article, we completed a function table with a linear function. We assumed a linear function and used the given points to find the slope and y-intercept. We then used the linear function to complete the function table.

However, we left the value of y as a variable. To find the value of y, we need more information about the function. We can use additional points on the function to find the value of y.

Conclusion

In conclusion, completing a function table requires understanding the relationship between the input values and the output values. We assumed a linear function and used the given points to find the slope and y-intercept. We then used the linear function to complete the function table.

Future Work

In future work, we can explore other types of functions, such as quadratic or exponential functions. We can also use more complex mathematical techniques, such as calculus, to analyze and complete function tables.

References

  • [1] "Functions" by Khan Academy
  • [2] "Linear Functions" by Math Is Fun
  • [3] "Completing a Function Table" by Purplemath
    Q&A: Completing a Function Table =====================================

Introduction

In our previous article, we explored a function table and completed it with a linear function. However, we left the value of y as a variable. In this article, we will answer some common questions related to completing a function table.

Q: What is a function table?

A function table is a table that shows the input values (x) and the corresponding output values (f(x)) of a function.

A: How do I complete a function table?

To complete a function table, you need to understand the relationship between the input values and the output values. You can use a linear function, quadratic function, or exponential function to complete the table.

Q: What is a linear function?

A linear function is a function that can be represented as:

f(x)=mx+bf(x) = mx + b

where m is the slope and b is the y-intercept.

A: How do I find the slope (m) and y-intercept (b)?

To find the slope (m) and y-intercept (b), you need at least two points on the function. You can use the slope formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1, y1) and (x2, y2) are the two points.

Q: What if I don't have two points on the function?

If you don't have two points on the function, you can use other methods to find the slope (m) and y-intercept (b). For example, you can use the point-slope form of a linear equation:

f(x)=m(x−x1)+y1f(x) = m(x - x_1) + y_1

Q: How do I use the completed function table?

Once you have completed the function table, you can use it to find the output values for different input values. You can also use it to graph the function.

Q: What are some common mistakes to avoid when completing a function table?

Some common mistakes to avoid when completing a function table include:

  • Not understanding the relationship between the input values and the output values
  • Not using the correct formula to find the slope (m) and y-intercept (b)
  • Not checking the work for errors

Q: How do I check my work for errors?

To check your work for errors, you can use the following steps:

  • Plug in the values of x and f(x) into the function to see if they match
  • Use a calculator or graphing tool to graph the function and check if it matches the completed function table
  • Check the work for algebraic errors, such as incorrect addition or subtraction

Conclusion

In conclusion, completing a function table requires understanding the relationship between the input values and the output values. You can use a linear function, quadratic function, or exponential function to complete the table. By following the steps outlined in this article, you can complete a function table and use it to find the output values for different input values.

Future Work

In future work, we can explore other types of functions, such as trigonometric or logarithmic functions. We can also use more complex mathematical techniques, such as calculus, to analyze and complete function tables.

References

  • [1] "Functions" by Khan Academy
  • [2] "Linear Functions" by Math Is Fun
  • [3] "Completing a Function Table" by Purplemath