Given The Function $f(x) = 2x - 7$, Solve For $f(x$\] When $x$ Is A Specific Value.

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 this article, we will focus on solving for a specific value in a linear function, using the function $f(x) = 2x - 7$ as an example.

Understanding Linear Functions

A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. The slope of a line represents the rate of change of the function, and it can be positive, negative, or zero. The y-intercept is the point where the line intersects the y-axis.

The Function $f(x) = 2x - 7$

The function $f(x) = 2x - 7$ is a linear function with a slope of 2 and a y-intercept of -7. This means that for every unit increase in $x$, the value of $f(x)$ increases by 2 units. The y-intercept of -7 means that when $x$ is equal to 0, the value of $f(x)$ is -7.

Solving for a Specific Value

To solve for a specific value in a linear function, we need to substitute the value of $x$ into the function and simplify. Let's say we want to find the value of $f(x)$ when $x$ is equal to 3.

Step 1: Substitute the value of $x$ into the function

$f(3) = 2(3) - 7$

Step 2: Simplify the expression

$f(3) = 6 - 7$

Step 3: Evaluate the expression

$f(3) = -1$

Therefore, the value of $f(x)$ when $x$ is equal to 3 is -1.

Example 2: Solving for a Specific Value

Let's say we want to find the value of $f(x)$ when $x$ is equal to -2.

Step 1: Substitute the value of $x$ into the function

$f(-2) = 2(-2) - 7$

Step 2: Simplify the expression

$f(-2) = -4 - 7$

Step 3: Evaluate the expression

$f(-2) = -11$

Therefore, the value of $f(x)$ when $x$ is equal to -2 is -11.

Conclusion

Solving for a specific value in a linear function is a straightforward process that involves substituting the value of $x$ into the function and simplifying the expression. By following the steps outlined in this article, you can easily solve for a specific value in a linear function, such as the function $f(x) = 2x - 7$.

Tips and Tricks

• Make sure to substitute the value of $x$ into the function correctly.
• Simplify the expression by combining like terms.
• Evaluate the expression to find the final answer.

Common Mistakes

• Failing to substitute the value of $x$ into the function correctly.
• Not simplifying the expression by combining like terms.
• Evaluating the expression incorrectly.

Real-World Applications

Solving for a specific value in a linear function has many real-world applications, such as:

• Calculating the cost of goods sold.
• Determining the amount of money earned from a sale.
• Finding the value of a stock or bond.

Final Thoughts

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Q: What is a linear function?

A: A linear function is a type of function that has a constant rate of change, and it can be represented by a linear equation. It is typically written in the form $f(x) = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Q: How do I solve for a specific value in a linear function?

A: To solve for a specific value in a linear function, you need to substitute the value of $x$ into the function and simplify the expression. This involves following the order of operations (PEMDAS) and combining like terms.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. It stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: What is the difference between a linear function and a quadratic function?

A: A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. A quadratic function, on the other hand, is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: Can I use a calculator to solve for a specific value in a linear function?

A: Yes, you can use a calculator to solve for a specific value in a linear function. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: What are some common mistakes to avoid when solving for a specific value in a linear function?

A: Some common mistakes to avoid when solving for a specific value in a linear function include:

• Failing to substitute the value of $x$ into the function correctly.
• Not simplifying the expression by combining like terms.
• Evaluating the expression incorrectly.

Q: How do I determine if a function is linear or not?

A: To determine if a function is linear or not, you need to check if it can be written in the form $f(x) = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. If it can be written in this form, then it is a linear function.

Q: Can I use a graphing calculator to graph a linear function?

A: Yes, you can use a graphing calculator to graph a linear function. This can be a useful tool for visualizing the function and understanding its behavior.

Q: What are some real-world applications of linear functions?

A: Some real-world applications of linear functions include:

• Calculating the cost of goods sold.
• Determining the amount of money earned from a sale.
• Finding the value of a stock or bond.

Q: Can I use linear functions to model real-world situations?

A: Yes, you can use linear functions to model real-world situations. For example, you can use a linear function to model the cost of goods sold, the amount of money earned from a sale, or the value of a stock or bond.

Q: How do I determine the slope of a linear function?

A: To determine the slope of a linear function, you need to look at the coefficient of the $x$ term. The coefficient of the $x$ term is the slope of the line.

Q: Can I use linear functions to solve optimization problems?

A: Yes, you can use linear functions to solve optimization problems. For example, you can use a linear function to find the maximum or minimum value of a function.

Q: How do I determine the y-intercept of a linear function?

A: To determine the y-intercept of a linear function, you need to look at the constant term. The constant term is the y-intercept of the line.