Evaluating a Function at a Given Value

Oct 13, 2025 by leolavanderia.com.br 19 views

Introduction

In mathematics, functions are used to describe relationships between variables. Evaluating a function at a given value means finding the output of the function when the input is equal to that value. In this article, we will focus on evaluating the function $g(x) = 3^x - 7$ at the value $x = 4$ . We will use this example to illustrate the steps involved in evaluating a function at a given value.

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 often denoted by a letter, such as $f(x)$ or $g(x)$ . The input is called the argument or the independent variable, and the output is called the value or the dependent variable.

Evaluating a Function at a Given Value

To evaluate a function at a given value, we need to substitute the value into the function and simplify the expression. In the case of the function $g(x) = 3^x - 7$ , we need to substitute $x = 4$ into the function.

Step 1: Substitute the Value into the Function

The first step is to substitute the value $x = 4$ into the function $g(x) = 3^x - 7$ . This means replacing $x$ with $4$ in the function.

g(4) = 3^4 - 7

Step 2: Simplify the Expression

The next step is to simplify the expression $3^4 - 7$ . To do this, we need to evaluate the exponentiation first.

3^4 = 3 \times 3 \times 3 \times 3 = 81

Now, we can substitute this value back into the expression.

g(4) = 81 - 7

Step 3: Evaluate the Expression

The final step is to evaluate the expression $81 - 7$ . This is a simple subtraction problem.

g(4) = 74

Conclusion

In this article, we evaluated the function $g(x) = 3^x - 7$ at the value $x = 4$ . We used the steps of substituting the value into the function, simplifying the expression, and evaluating the expression to find the output of the function. The final answer is $g(4) = 74$ .

Example Problems

Here are a few example problems to help you practice evaluating functions at given values.

Example 1

Evaluate the function $f(x) = 2x + 5$ at the value $x = 3$ .

Step 1: Substitute the Value into the Function

f(3) = 2(3) + 5

Step 2: Simplify the Expression

f(3) = 6 + 5

Step 3: Evaluate the Expression

f(3) = 11

Example 2

Evaluate the function $h(x) = x^2 - 4$ at the value $x = 2$ .

Step 1: Substitute the Value into the Function

h(2) = (2)^2 - 4

Step 2: Simplify the Expression

h(2) = 4 - 4

Step 3: Evaluate the Expression

h(2) = 0

Tips and Tricks

Here are a few tips and tricks to help you evaluate functions at given values.

Make sure to substitute the value into the function correctly.
Simplify the expression as much as possible before evaluating it.
Use the order of operations (PEMDAS) to evaluate the expression correctly.
Check your work by plugging the value back into the function and simplifying the expression.

Common Mistakes

Here are a few common mistakes to avoid when evaluating functions at given values.

Not substituting the value into the function correctly.
Not simplifying the expression enough before evaluating it.
Not using the order of operations (PEMDAS) to evaluate the expression correctly.
Not checking your work by plugging the value back into the function and simplifying the expression.

Conclusion

Introduction

In our previous article, we discussed how to evaluate functions at given values. In this article, we will answer some common questions related to evaluating functions at given values.

Q: What is the difference between evaluating a function at a given value and finding the value of a function?

A: Evaluating a function at a given value means finding the output of the function when the input is equal to that value. Finding the value of a function means finding the value of the function for a specific input.

Q: How do I know if I have evaluated a function correctly?

A: To check if you have evaluated a function correctly, plug the value back into the function and simplify the expression. If the result is the same as the original expression, then you have evaluated the function correctly.

Q: What if I get a different answer when I plug the value back into the function?

A: If you get a different answer when you plug the value back into the function, then you made a mistake when you evaluated the function. Go back and re-evaluate the function to find the correct answer.

Q: Can I evaluate a function at a given value if the function is not defined at that value?

A: No, you cannot evaluate a function at a given value if the function is not defined at that value. In this case, the function is said to be undefined at that value.

Q: How do I know if a function is defined at a given value?

A: To check if a function is defined at a given value, look at the function's domain. If the given value is in the domain, then the function is defined at that value.

Q: What if I get a complex expression when I evaluate a function at a given value?

A: If you get a complex expression when you evaluate a function at a given value, simplify the expression as much as possible before evaluating it. You can use algebraic manipulations, such as combining like terms and factoring, to simplify the expression.

Q: Can I use a calculator to evaluate a function at a given value?

A: Yes, you can use a calculator to evaluate a function at a given value. However, make sure to check your work by plugging the value back into the function and simplifying the expression.

Q: How do I know if I have evaluated a function correctly using a calculator?

A: To check if you have evaluated a function correctly using a calculator, plug the value back into the function and simplify the expression. If the result is the same as the original expression, then you have evaluated the function correctly.

Q: What if I get a different answer when I use a calculator to evaluate a function at a given value?

A: If you get a different answer when you use a calculator to evaluate a function at a given value, then you made a mistake when you entered the function or the value into the calculator. Go back and re-enter the function and the value into the calculator to find the correct answer.

Conclusion

Evaluating functions at given values is an important concept in mathematics. By following the steps of substituting the value into the function, simplifying the expression, and evaluating the expression, you can find the output of the function. Remember to check your work by plugging the value back into the function and simplifying the expression. With practice, you will become more comfortable evaluating functions at given values.

Common Mistakes

Here are a few common mistakes to avoid when evaluating functions at given values.

Not substituting the value into the function correctly.
Not simplifying the expression enough before evaluating it.
Not using the order of operations (PEMDAS) to evaluate the expression correctly.
Not checking your work by plugging the value back into the function and simplifying the expression.
Using a calculator incorrectly.

Tips and Tricks

Here are a few tips and tricks to help you evaluate functions at given values.

Make sure to substitute the value into the function correctly.
Simplify the expression as much as possible before evaluating it.
Use the order of operations (PEMDAS) to evaluate the expression correctly.
Check your work by plugging the value back into the function and simplifying the expression.
Use a calculator to evaluate a function at a given value, but make sure to check your work.