Evaluating F(x) = 2x - 1 For Various Inputs

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This article provides a comprehensive guide on how to evaluate the function f(x) = 2x - 1 for different input values. We will explore the process step-by-step, providing clear explanations and examples to enhance your understanding. Whether you're a student learning about functions or simply looking to brush up on your math skills, this guide will equip you with the knowledge to confidently tackle function evaluations. Let's dive into the world of functions and discover how to find the values of f(x) = 2x - 1 for various inputs.

(1) Evaluating f(3)

Understanding the Basics of Function Evaluation

At the heart of function evaluation lies the concept of substitution. To evaluate f(3), we replace the variable x in the function's expression with the numerical value 3. This straightforward substitution forms the cornerstone of function evaluation, allowing us to determine the output of the function for a specific input.

Step-by-Step Evaluation of f(3)

  1. Write down the function: Begin by clearly stating the function you're working with, which in this case is f(x) = 2x - 1. This serves as your foundation for the evaluation process.
  2. Substitute x with 3: Replace every instance of the variable x in the function's expression with the value 3. This substitution is the crux of function evaluation, as it connects the input to the output.
  3. Simplify the expression: After the substitution, you'll have a numerical expression. Perform the necessary arithmetic operations, following the order of operations (PEMDAS/BODMAS), to simplify the expression and arrive at the final value.

Detailed Calculation

f(3) = 2(3) - 1

Here, we've replaced x with 3.

Next, we perform the multiplication:

f(3) = 6 - 1

Finally, we subtract:

f(3) = 5

Therefore, the value of the function f(x) = 2x - 1 when x is 3 is 5. This signifies that when the input to the function is 3, the corresponding output is 5.

Visualizing the Result

We can visualize this result graphically. The function f(x) = 2x - 1 represents a straight line. The point (3, 5) lies on this line, indicating that when x is 3, the y-value (which represents f(x)) is 5. This graphical representation provides a visual confirmation of our calculated result.

(2) Evaluating f(-1)

Extending the Concept to Negative Inputs

Function evaluation isn't limited to positive inputs. We can readily evaluate functions for negative values as well. The process remains the same: substitute the variable x with the given negative value and simplify the resulting expression. This adaptability to various input types makes function evaluation a versatile tool in mathematics.

Step-by-Step Evaluation of f(-1)

  1. Write down the function: As before, start by stating the function: f(x) = 2x - 1.
  2. Substitute x with -1: Replace x with -1 in the function's expression. Remember to pay close attention to signs when dealing with negative numbers.
  3. Simplify the expression: Perform the arithmetic operations, adhering to the order of operations, to simplify the expression and obtain the final value.

Detailed Calculation

f(-1) = 2(-1) - 1

We've substituted x with -1.

Now, perform the multiplication:

f(-1) = -2 - 1

Finally, subtract:

f(-1) = -3

Thus, the value of the function f(x) = 2x - 1 when x is -1 is -3. This signifies that for an input of -1, the function produces an output of -3.

Interpreting the Negative Result

The negative result, f(-1) = -3, simply indicates that the corresponding point on the graph of the function lies below the x-axis. The graph of f(x) = 2x - 1 is a straight line, and the point (-1, -3) is located in the third quadrant of the coordinate plane. This negative output is a natural consequence of the function's definition and the negative input value.

(3) Evaluating f(a²)

Introducing Algebraic Expressions as Inputs

Function evaluation isn't confined to numerical inputs alone. We can also substitute algebraic expressions, such as a², for the variable x. This extends the applicability of function evaluation, allowing us to explore the function's behavior for a range of values represented by the expression.

Step-by-Step Evaluation of f(a²)

  1. State the function: Begin by stating the function: f(x) = 2x - 1.
  2. Substitute x with a²: Replace x with the algebraic expression a². This substitution is analogous to numerical substitution, but it results in an algebraic expression rather than a numerical value.
  3. Simplify the expression: Perform any possible simplifications. In this case, the only simplification is the multiplication of 2 and a².

Detailed Calculation

f(a²) = 2(a²) - 1

Here, we've replaced x with a².

Now, perform the multiplication:

f(a²) = 2a² - 1

This expression, 2a² - 1, represents the value of the function f(x) = 2x - 1 when x is a². It's an algebraic expression that depends on the value of a. This outcome highlights the power of function evaluation to generate algebraic expressions as outputs.

Understanding the Resulting Expression

The expression 2a² - 1 represents a family of values, each corresponding to a different value of a. For instance, if a is 2, then f(a²) = 2(2²) - 1 = 7. This demonstrates that evaluating a function with an algebraic expression as input yields another algebraic expression that captures the function's behavior for a range of input values. This concept is crucial in various mathematical contexts, including calculus and algebra.

(4) Evaluating f(a - 1)

Working with More Complex Algebraic Inputs

Function evaluation can handle even more intricate algebraic expressions as inputs. Consider evaluating f(a - 1). This involves substituting the binomial expression (a - 1) for the variable x. The process remains consistent, but the simplification step might require additional algebraic manipulations.

Step-by-Step Evaluation of f(a - 1)

  1. State the function: Begin by stating the function: f(x) = 2x - 1.
  2. Substitute x with (a - 1): Replace x with the binomial expression (a - 1). This substitution is a direct application of the function evaluation principle.
  3. Simplify the expression: This step involves distributing the 2 and then combining like terms.

Detailed Calculation

f(a - 1) = 2(a - 1) - 1

We've substituted x with (a - 1).

Now, distribute the 2:

f(a - 1) = 2a - 2 - 1

Finally, combine the constant terms:

f(a - 1) = 2a - 3

The resulting expression, 2a - 3, represents the value of the function f(x) = 2x - 1 when x is (a - 1). This expression is linear in a, indicating a linear relationship between a and the function's output for this specific input. This example showcases the versatility of function evaluation in dealing with various algebraic inputs.

Interpreting the Linear Expression

The expression 2a - 3 represents a straight line when plotted as a function of a. The slope of this line is 2, and the y-intercept is -3. This linear relationship provides insights into how the function f(x) = 2x - 1 behaves when its input is shifted by 1 (x = a - 1). Understanding these relationships is crucial for analyzing functions and their transformations.

Conclusion

In this comprehensive guide, we've explored the process of evaluating the function f(x) = 2x - 1 for various inputs, including numerical values, squared terms, and binomial expressions. We've demonstrated how the core principle of substitution, coupled with algebraic simplification, allows us to determine the function's output for a wide range of inputs. This understanding is fundamental to grasping the concept of functions and their applications in mathematics and beyond. By mastering function evaluation, you'll be well-equipped to tackle more advanced mathematical concepts and problem-solving scenarios.