Given $f(x) = 10 - 2x$, Find $f(7)$.A. -4 B. 3 C. 7 D. 56
Introduction
In mathematics, functions are used to describe the relationship between variables. A linear function is a type of function that has a constant rate of change, and it can be represented by a linear equation in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. In this article, we will focus on evaluating a linear function, specifically the function f(x) = 10 - 2x, at a given value of x.
Understanding the Function
The given function is f(x) = 10 - 2x. This is a linear function with a slope of -2 and a y-intercept of 10. The slope represents the rate of change of the function, and the y-intercept represents the point where the function intersects the y-axis.
Evaluating the Function at x = 7
To evaluate the function f(x) = 10 - 2x at x = 7, we need to substitute x = 7 into the function and simplify the expression.
Step 1: Substitute x = 7 into the function
f(7) = 10 - 2(7)
Step 2: Simplify the expression
f(7) = 10 - 14
Step 3: Evaluate the expression
f(7) = -4
Therefore, the value of the function f(x) = 10 - 2x at x = 7 is -4.
Conclusion
In this article, we have evaluated the linear function f(x) = 10 - 2x at x = 7. We have followed a step-by-step approach to substitute x = 7 into the function and simplify the expression. The final answer is -4.
Key Takeaways
- A linear function is a type of function that has a constant rate of change.
- The slope of a linear function represents the rate of change of the function.
- The y-intercept of a linear function represents the point where the function intersects the y-axis.
- To evaluate a linear function at a given value of x, we need to substitute x into the function and simplify the expression.
Practice Problems
- Evaluate the function f(x) = 3x + 2 at x = 5.
- Evaluate the function f(x) = 2x - 5 at x = 3.
- Evaluate the function f(x) = x^2 + 1 at x = 4.
Solutions
- f(5) = 3(5) + 2 = 17
- f(3) = 2(3) - 5 = -1
- f(4) = (4)^2 + 1 = 17
Final Thoughts
Introduction
In our previous article, we discussed how to evaluate a linear function at a given value of x. In this article, we will provide a Q&A guide to help you understand and apply the concepts of evaluating linear functions.
Q: What is a linear function?
A: A linear function is a type of function that has a constant rate of change. It can be represented by a linear equation in the form of f(x) = mx + b, where m is the slope and b is the y-intercept.
Q: What is the slope of a linear function?
A: The slope of a linear function represents the rate of change of the function. It is a measure of how much the function changes for a one-unit change in x.
Q: What is the y-intercept of a linear function?
A: The y-intercept of a linear function represents the point where the function intersects the y-axis. It is the value of the function when x = 0.
Q: How do I evaluate a linear function at a given value of x?
A: To evaluate a linear function at a given value of x, you need to substitute x into the function and simplify the expression. For example, if we have the function f(x) = 2x + 3 and we want to evaluate it at x = 4, we would substitute x = 4 into the function and simplify the expression:
f(4) = 2(4) + 3 f(4) = 8 + 3 f(4) = 11
Q: What if the function has a negative slope?
A: If the function has a negative slope, it means that the function decreases as x increases. For example, if we have the function f(x) = -2x + 5 and we want to evaluate it at x = 3, we would substitute x = 3 into the function and simplify the expression:
f(3) = -2(3) + 5 f(3) = -6 + 5 f(3) = -1
Q: Can I evaluate a linear function at a fraction or decimal value of x?
A: Yes, you can evaluate a linear function at a fraction or decimal value of x. For example, if we have the function f(x) = 3x + 2 and we want to evaluate it at x = 2.5, we would substitute x = 2.5 into the function and simplify the expression:
f(2.5) = 3(2.5) + 2 f(2.5) = 7.5 + 2 f(2.5) = 9.5
Q: What if the function has a variable in the exponent?
A: If the function has a variable in the exponent, it is not a linear function. For example, if we have the function f(x) = 2x^2 + 3, it is not a linear function because the exponent is 2, not 1.
Q: Can I use a calculator to evaluate a linear function?
A: Yes, you can use a calculator to evaluate a linear function. Simply enter the function and the value of x into the calculator and press the "enter" or "=" button to get the result.
Conclusion
Evaluating linear functions is an essential skill in mathematics, and it has numerous applications in real-life situations. By following the steps outlined in this Q&A guide, you can easily evaluate linear functions at given values of x.