Consider The Function Represented By The Equation $6q = 3s - 9$. Write The Equation In Function Notation, Where $q$ Is The Independent Variable.A. $f(a) = \frac{1}{2}q - \frac{3}{2}$B. $f(c) = 2s + 3$C. $f(s) =

Introduction

In mathematics, function notation is a way of representing a function using a specific notation. It is a powerful tool for expressing and working with functions in a concise and elegant way. In this article, we will explore how to write an equation in function notation, where the variable $q$ is the independent variable.

Understanding the Equation

The given equation is $6q = 3s - 9$. To write this equation in function notation, we need to isolate the variable $q$ and express it in terms of the variable $s$. This will allow us to represent the function as a relation between the input variable $s$ and the output variable $q$.

Isolating the Variable q

To isolate the variable $q$, we can start by dividing both sides of the equation by 6. This will give us:

$$q = \frac{3s - 9}{6}$$

Simplifying the Expression

We can simplify the expression by dividing the numerator and denominator by their greatest common divisor, which is 3. This will give us:

$$q = \frac{s - 3}{2}$$

Writing the Equation in Function Notation

Now that we have isolated the variable $q$ and simplified the expression, we can write the equation in function notation. The function notation is represented as $f(x) = y$, where $x$ is the input variable and $y$ is the output variable.

In this case, the input variable is $s$ and the output variable is $q$. Therefore, we can write the equation in function notation as:

$$f(s) = \frac{s - 3}{2}$$

Conclusion

In this article, we have seen how to write an equation in function notation, where the variable $q$ is the independent variable. We started by isolating the variable $q$ and simplifying the expression. Then, we wrote the equation in function notation, using the input variable $s$ and the output variable $q$. The final answer is $f(s) = \frac{s - 3}{2}$.

Discussion

The equation $6q = 3s - 9$ can be written in function notation as $f(s) = \frac{s - 3}{2}$. This represents a linear function, where the output variable $q$ is a linear function of the input variable $s$.

The function $f(s) = \frac{s - 3}{2}$ has a slope of $\frac{1}{2}$ and a y-intercept of $-\frac{3}{2}$. This means that for every unit increase in the input variable $s$, the output variable $q$ increases by $\frac{1}{2}$ unit.

Example

Suppose we want to find the value of $q$ when $s = 5$. We can plug in the value of $s$ into the function $f(s) = \frac{s - 3}{2}$ and solve for $q$.

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

Therefore, when $s = 5$, the value of $q$ is 1.

Exercise

Write the equation $4x = 2y - 6$ in function notation, where the variable $x$ is the independent variable.

Answer

To write the equation in function notation, we need to isolate the variable $x$ and express it in terms of the variable $y$. This will give us:

$$x = \frac{2y - 6}{4} = \frac{y - 3}{2}$$

Therefore, the equation in function notation is $f(y) = \frac{y - 3}{2}$.

Conclusion

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Introduction

In our previous article, we explored how to write an equation in function notation, where the variable $q$ is the independent variable. We also saw how to simplify the expression and write the equation in function notation. In this article, we will answer some frequently asked questions about function notation and linear functions.

Q: What is function notation?

A: Function notation is a way of representing a function using a specific notation. It is a powerful tool for expressing and working with functions in a concise and elegant way. In function notation, the input variable is represented as $x$ and the output variable is represented as $f(x)$.

Q: How do I write an equation in function notation?

A: To write an equation in function notation, you need to isolate the variable and express it in terms of the input variable. This will give you the function notation of the equation.

Q: What is a linear function?

A: A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Linear functions have a constant rate of change and can be represented graphically as a straight line.

Q: How do I find the slope and y-intercept of a linear function?

A: To find the slope and y-intercept of a linear function, you need to look at the equation in function notation. The slope is the coefficient of the input variable, and the y-intercept is the constant term.

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

A: A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. A non-linear function is a function that cannot be written in this form. Non-linear functions have a variable rate of change and can be represented graphically as a curve.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot two points on the graph and draw a straight line through them. You can also use the slope and y-intercept to find the equation of the line.

Q: What is the domain and range of a linear function?

A: The domain of a linear function is the set of all possible input values, and the range is the set of all possible output values. For a linear function, the domain and range are all real numbers.

Q: How do I find the inverse of a linear function?

A: To find the inverse of a linear function, you need to swap the input and output variables and solve for the new input variable.

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

A: A function is a relation where each input value corresponds to exactly one output value. A relation is a set of ordered pairs, where each input value may correspond to more than one output value.

Conclusion

In this article, we have answered some frequently asked questions about function notation and linear functions. We have seen how to write an equation in function notation, how to find the slope and y-intercept of a linear function, and how to graph a linear function. We have also seen the difference between a linear function and a non-linear function, and how to find the inverse of a linear function.

Practice Problems

1. Write the equation $2x = 3y - 6$ in function notation, where the variable $x$ is the independent variable.
2. Find the slope and y-intercept of the linear function $f(x) = 2x + 3$.
3. Graph the linear function $f(x) = 2x - 1$.
4. Find the inverse of the linear function $f(x) = 2x + 3$.
5. What is the domain and range of the linear function $f(x) = 2x + 3$?

Answers

1. $f(y) = \frac{3y - 6}{2}$
2. Slope: 2, y-intercept: 3
3. Graph: a straight line with a slope of 2 and a y-intercept of -1
4. $f^{-1}(x) = \frac{x - 3}{2}$
5. Domain: all real numbers, range: all real numbers