Perform the necessary calculations

Oct 13, 2025 by miodeya.com 13 views

Introduction

In mathematics, functions are a fundamental concept that helps us describe relationships between variables. Evaluating a function at a specific value of its variable is a crucial aspect of mathematics, and it has numerous applications in various fields, including science, engineering, and economics. In this article, we will focus on evaluating the function $f(x) = \frac{1}{2}x + 3$ at $x = 4$ .

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a rule that assigns to each input exactly one output. In other words, for every input, there is a corresponding output. The function is often denoted by a symbol, such as $f(x)$ , where $x$ is the input variable.

Evaluating a Function

To evaluate a function at a specific value of its variable, we simply substitute the value into the function and perform the necessary calculations. In the case of the function $f(x) = \frac{1}{2}x + 3$ , we need to substitute $x = 4$ into the function.

Step-by-Step Evaluation

Substitute the value of x: We substitute $x = 4$ into the function $f(x) = \frac{1}{2}x + 3$ .
Perform the necessary calculations: We multiply $\frac{1}{2}$ by $4$ and add $3$ to get the final result.

Calculations

$f(4) = \frac{1}{2}(4) + 3$ $f(4) = 2 + 3$ $f(4) = 5$

Conclusion

In conclusion, evaluating a function at a specific value of its variable is a straightforward process that involves substituting the value into the function and performing the necessary calculations. In this article, we evaluated the function $f(x) = \frac{1}{2}x + 3$ at $x = 4$ and obtained the result $f(4) = 5$ . This result is a specific output of the function for a given input value.

Answer

The correct answer is a. 5.

Additional Examples

To further illustrate the concept of evaluating a function, let's consider a few more examples.

Example 1

Evaluate the function $f(x) = 2x - 1$ at $x = 3$ .

Step-by-Step Evaluation

Substitute the value of x: We substitute $x = 3$ into the function $f(x) = 2x - 1$ .
Perform the necessary calculations: We multiply $2$ by $3$ and subtract $1$ to get the final result.

Calculations

$f(3) = 2(3) - 1$ $f(3) = 6 - 1$ $f(3) = 5$

Example 2

Evaluate the function $f(x) = \frac{1}{3}x + 2$ at $x = 6$ .

Step-by-Step Evaluation

Substitute the value of x: We substitute $x = 6$ into the function $f(x) = \frac{1}{3}x + 2$ .
Perform the necessary calculations: We multiply $\frac{1}{3}$ by $6$ and add $2$ to get the final result.

Calculations

$f(6) = \frac{1}{3}(6) + 2$ $f(6) = 2 + 2$ $f(6) = 4$

Example 3

Evaluate the function $f(x) = x^2 + 1$ at $x = 2$ .

Step-by-Step Evaluation

Substitute the value of x: We substitute $x = 2$ into the function $f(x) = x^2 + 1$ .
Perform the necessary calculations: We square $2$ and add $1$ to get the final result.

Calculations

$f(2) = (2)^2 + 1$ $f(2) = 4 + 1$ $f(2) = 5$

Conclusion

Introduction

In our previous article, we discussed the concept of evaluating functions at specific values of their variables. We provided step-by-step examples and calculations to illustrate the process. In this article, we will answer some frequently asked questions (FAQs) related to evaluating functions.

Q&A

Q1: What is the difference between evaluating a function and finding the value of a function?

A1: Evaluating a function at a specific value of its variable means finding the output of the function for that particular input value. Finding the value of a function, on the other hand, means finding the general expression for the function.

Q2: How do I know if a function is defined at a specific value of its variable?

A2: A function is defined at a specific value of its variable if the function is continuous at that point and the input value is within the domain of the function.

Q3: What is the difference between a function and a relation?

A3: A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input corresponds to exactly one output. A relation, on the other hand, is a set of ordered pairs where each input may correspond to multiple outputs.

Q4: Can a function have multiple outputs for a single input value?

A4: No, a function cannot have multiple outputs for a single input value. By definition, a function assigns to each input exactly one output.

Q5: How do I evaluate a function with multiple variables?

A5: To evaluate a function with multiple variables, you need to substitute the values of all the variables into the function and perform the necessary calculations.

Q6: Can I evaluate a function at a value that is not in its domain?

A6: No, you cannot evaluate a function at a value that is not in its domain. The function is not defined at such values, and the evaluation process will not yield a valid result.

Q7: How do I know if a function is linear or nonlinear?

A7: A function is linear if it can be expressed in the form $f(x) = mx + b$ , where $m$ and $b$ are constants. A function is nonlinear if it cannot be expressed in this form.

Q8: Can I evaluate a function at a complex number?

A8: Yes, you can evaluate a function at a complex number. However, the evaluation process may involve complex arithmetic and may not yield a real number.

Q9: How do I evaluate a function with a trigonometric or exponential term?

A9: To evaluate a function with a trigonometric or exponential term, you need to use the properties of trigonometric and exponential functions, such as the Pythagorean identity and the exponential function's properties.

Q10: Can I evaluate a function at a value that is not in its range?

A10: No, you cannot evaluate a function at a value that is not in its range. The function is not defined at such values, and the evaluation process will not yield a valid result.

Conclusion

In conclusion, evaluating functions is a crucial aspect of mathematics that has numerous applications in various fields. By understanding the concepts and techniques discussed in this article, you can evaluate functions with confidence and accuracy. Remember to always follow the step-by-step evaluation process and to check if the function is defined at the specific value of its variable.

Additional Resources

For further learning and practice, we recommend the following resources:

Khan Academy: Functions and Relations
MIT OpenCourseWare: Calculus
Wolfram Alpha: Function Evaluation

Practice Problems

To practice evaluating functions, try the following problems:

Evaluate the function $f(x) = 2x - 1$ at $x = 4$ .
Evaluate the function $f(x) = \frac{1}{3}x + 2$ at $x = 6$ .
Evaluate the function $f(x) = x^2 + 1$ at $x = 2$ .
Evaluate the function $f(x) = \sin x$ at $x = \frac{\pi}{2}$ .
Evaluate the function $f(x) = e^x$ at $x = 1$ .

Answer Key

$f(4) = 7$
$f(6) = 4$
$f(2) = 5$
$f(\frac{\pi}{2}) = 1$
$f(1) = e$