Understanding the Function F(x) = 9x - 10
If F(x) = 9x - 10 find F(3)= is a basic algebraic problem that involves evaluating a linear function at a specific value of x. To solve this problem, it's essential to understand the structure of the function F(x), how it operates, and the general process of plugging in a value for x to find the corresponding output. This type of problem is fundamental in algebra and serves as a foundation for more complex functions and mathematical concepts.
Breaking Down the Function F(x) = 9x - 10
The Nature of the Function
The function F(x) = 9x - 10 is a linear function, which means its graph is a straight line. The general form of a linear function is:
- F(x) = mx + b
where:
- m is the slope of the line, indicating how steep it is.
- b is the y-intercept, the point where the line crosses the y-axis.
In this specific case, the slope m is 9, and the y-intercept b is -10, indicating that the line rises quickly due to the large slope and crosses the y-axis at -10.
Implications of the Function
The function describes a relationship where the output F(x) depends linearly on the input x. As x increases or decreases, F(x) changes proportionally, scaled by the slope 9, and shifted vertically by -10.
Understanding this relationship allows us to predict the value of F(x) for any given x, including the specific value x=3, which is the focus of our problem.
Calculating F(3)
Step-by-step Evaluation
To find F(3), substitute x=3 into the function:
F(3) = 9(3) - 10
Next, perform the multiplication:
F(3) = 27 - 10
Finally, perform the subtraction:
F(3) = 17
Thus, the value of the function at x=3 is 17.
Verification of the Calculation
It's always good practice to verify the calculation. Double-checking:
- Multiply the input by the slope: 9 × 3 = 27.
- Subtract the y-intercept: 27 - 10 = 17.
The calculation confirms that F(3) = 17.
Understanding the Significance of F(3)=17
Interpretation in Context
While the problem is purely algebraic, understanding what F(3)=17 signifies can be insightful. For example, if F(x) represented a real-world quantity, such as the total cost based on the number of items (x), then:
- When x=3 (say, 3 items), the total cost would be 17 units of currency.
This illustrates the importance of evaluating functions at specific points to determine real-world outcomes.
Graphical Representation
Plotting the function F(x) = 9x - 10 on a graph helps visualize how the function behaves. Key points include:
- The y-intercept at (0, -10).
- The point (3, 17), which we calculated.
The line passing through these points demonstrates the linear relationship, with a steep slope of 9, indicating rapid change.
Broader Applications of Linear Functions
Real-World Examples
Linear functions like F(x) = 9x - 10 are used extensively in various fields:
- Economics: Calculating profit based on units sold.
- Physics: Determining distance traveled over time with constant speed.
- Statistics: Modeling relationships between variables.
Mathematical Significance
Understanding how to evaluate functions at specific points is fundamental in mathematics. It helps in:
- Analyzing the behavior of functions.
- Graphing functions accurately.
- Understanding the relationship between variables.
Practice Problems and Further Exploration
Additional Evaluations
To deepen understanding, consider evaluating F(x) at different values:
- F(0): What is the output when x=0?
- F(5): What about x=5?
- F(-2): How does the function behave for negative x?
Sample Solutions
- F(0) = 9(0) - 10 = -10
- F(5) = 9(5) - 10 = 45 - 10 = 35
- F(-2) = 9(-2) - 10 = -18 - 10 = -28
Conclusion
In summary, evaluating the function F(x) = 9x - 10 at x=3 involves straightforward substitution and arithmetic. The result, F(3) = 17, exemplifies the linear relationship defined by the function. This process underscores fundamental algebraic skills and highlights the importance of understanding function behavior, both in mathematical theory and real-world applications. Mastery of such evaluations forms the basis for more advanced studies in algebra, calculus, and applied mathematics.
