Function Notation: Expressing 9x + 3y = 12 As F(x)

Hey guys! Let's dive into how to express the function represented by the equation $9x + 3y = 12$ using function notation, with $x$ as the independent variable. This is a common task in algebra, and understanding how to do it is super useful for all sorts of mathematical problems. So, let's break it down step by step to make sure we get it right.

Understanding Function Notation

First, let's quickly recap what function notation actually means. Function notation is a way of writing equations that clearly shows the input and output relationship. Instead of writing something like $y = mx + b$, we write $f(x) = mx + b$. Here, $f(x)$ is read as "f of x," and it represents the value of the function $f$ when the input is $x$. The independent variable is $x$, and the dependent variable is $f(x)$, which is the same as $y$. Basically, function notation tells us what the function does to the input variable to produce the output.

When you're dealing with function notation, it’s all about isolating the dependent variable (usually y) in terms of the independent variable (usually x). This allows you to clearly see how the output of the function changes as the input x varies. This is particularly useful in calculus, where you often need to find the derivative of a function, or in graphing, where you want to plot the function on a coordinate plane. Always remember, function notation is a powerful tool that offers clarity and precision in mathematical expressions. So, make sure you feel comfortable manipulating equations into this form.

Why bother with function notation anyway? Well, it makes things clearer when you have multiple functions. For example, you can have $f(x)$, $g(x)$, and $h(x)$, and it's easy to see which function you're talking about. It also helps when you're evaluating functions for specific values of $x$. For instance, if $f(x) = 2x + 3$, then $f(5)$ means you substitute $x = 5$ into the equation, giving you $f(5) = 2(5) + 3 = 13$. Function notation is super handy for both theoretical math and practical applications!

Converting the Equation to Function Notation

Now, let's get to the equation we're working with: $9x + 3y = 12$. Our goal is to isolate $y$ on one side of the equation. Here’s how we can do it:

1. Subtract $9x$ from both sides: $3y = -9x + 12$

2. Divide both sides by 3: $y = -3x + 4$

Now that we have $y$ isolated, we can replace $y$ with $f(x)$ to write the equation in function notation:

$f(x) = -3x + 4$

And that's it! We've successfully expressed the given equation in function notation with $x$ as the independent variable.

Detailed Explanation of Each Step

Let's walk through each step a little more slowly to make sure everyone's on the same page. First, we start with the equation $9x + 3y = 12$. The objective is to get $y$ by itself on one side of the equation. To do this, we need to undo the operations that are being applied to $y$.

The first operation we address is the addition of $9x$. To undo this, we subtract $9x$ from both sides of the equation. This keeps the equation balanced and isolates the term with $y$: $3y = -9x + 12$.

Next, we see that $y$ is being multiplied by 3. To undo this multiplication, we divide both sides of the equation by 3. Again, this maintains the balance of the equation and solves for $y$: $y = -3x + 4$.

Finally, we replace $y$ with $f(x)$ to express the equation in function notation. So, we have $f(x) = -3x + 4$. This tells us that the function $f$ takes an input $x$, multiplies it by -3, and then adds 4 to get the output. This is a clear and concise way to represent the relationship between $x$ and $y$.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided and see which one matches our result:

A. $f(x) = -"1/3 y + 4/3$ B. $f(x) = -3x + 4$ C. $f(x) = -"1/3 x + 4/3$ D. $m(y) = -3y + 4$

• Option A is incorrect because it still includes $y$ in the function of $x$, which doesn't make sense.
• Option B is exactly what we found: $f(x) = -3x + 4$. So, this is the correct answer!
• Option C is similar to option A but has $x$ instead of $y$. Still, it doesn't match our derived function.
• Option D is a function of $y$, $m(y)$, which is not what we were asked to find. We wanted a function of $x$.

Therefore, the correct answer is B. $f(x) = -3x + 4$.

Common Mistakes to Avoid

When working with function notation and converting equations, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

Mistake 1: Forgetting to Distribute

One common mistake is forgetting to distribute when dividing or multiplying. For example, if you have $2(x + 3) = 8$, you need to distribute the 2 to both the $x$ and the 3, resulting in $2x + 6 = 8$. Failing to do this will lead to an incorrect answer. Always double-check that you've distributed correctly.

Mistake 2: Incorrectly Combining Like Terms

Another frequent error is incorrectly combining like terms. For instance, $3x + 2y$ cannot be combined because $3x$ and $2y$ are not like terms. Only terms with the same variable and exponent can be combined. Make sure you are only combining terms that are actually like terms.

Mistake 3: Not Isolating the Correct Variable

In the context of function notation, a critical mistake is not isolating the correct variable. When asked to express an equation as $f(x)$, you must isolate $y$ (or $f(x)$) in terms of $x$. If you accidentally isolate $x$ in terms of $y$, you won't have the function in the correct form. Always double-check which variable needs to be isolated based on the function notation you're aiming for.

Mistake 4: Confusing f(x) with Multiplication

Some students mistakenly think that $f(x)$ means $f$ times $x$. Remember, $f(x)$ is a notation that represents the value of the function $f$ at the input $x$. It's not multiplication; it's a way of expressing the output of a function for a given input. Understanding this distinction is crucial for correctly interpreting and using function notation.

Mistake 5: Sign Errors

Sign errors are incredibly common and can easily throw off your answer. When moving terms from one side of the equation to the other, remember to change the sign. For example, if you have $x + 5 = 10$, subtracting 5 from both sides gives you $x = 10 - 5$, not $x = 10 + 5$. Pay close attention to the signs when rearranging equations.

Conclusion

So, to wrap it up, expressing the function $9x + 3y = 12$ in function notation with $x$ as the independent variable gives us $f(x) = -3x + 4$. Remember to isolate $y$, and you're golden! Keep practicing, and you'll nail this concept in no time. You got this!