Math  /  Algebra

QuestionWhich of the following is the correct simplified difference quotient for f(x)=2xx3?f(x)=\frac{2 x}{x-3} ? 2(x+h3)(x3)-\frac{2}{(x+h-3)(x-3)} 2(x+h3)(x3)\frac{2}{(x+h-3)(x-3)} 2h(x+h3)(x3)-\frac{2 h}{(x+h-3)(x-3)} 2h(x+h3)(x3)\frac{2 h}{(x+h-3)(x-3)} Nothing in this list is correct.

Studdy Solution

STEP 1

What is this asking? We need to find the simplified difference quotient for the given function f(x)f(x). Watch out! Don't forget to distribute the negative sign correctly when simplifying the difference quotient!

STEP 2

1. Define the Difference Quotient
2. Substitute and Simplify
3. Find a Common Denominator
4. Simplify the Numerator
5. Divide Out hh

STEP 3

Alright, let's **kick things off** by remembering what the difference quotient is!
It's a way to find the **average rate of change** of a function over a tiny interval.
The formula is: f(x+h)f(x)h \frac{f(x+h) - f(x)}{h} This represents the change in the function's value divided by the change in xx, which is just hh.

STEP 4

Now, let's **plug in** our function f(x)=2xx3f(x) = \frac{2x}{x-3} into the difference quotient formula: 2(x+h)(x+h)32xx3h \frac{\frac{2(x+h)}{(x+h)-3} - \frac{2x}{x-3}}{h} So, f(x+h)f(x+h) means we replace every xx in our function with x+hx+h.

STEP 5

To subtract those fractions in the numerator, we need a **common denominator**.
Let's multiply the first fraction by x3x3\frac{x-3}{x-3} and the second fraction by x+h3x+h3\frac{x+h-3}{x+h-3}.
This gives us: 2(x+h)(x3)(x+h3)(x3)2x(x+h3)(x+h3)(x3)h \frac{\frac{2(x+h)(x-3)}{(x+h-3)(x-3)} - \frac{2x(x+h-3)}{(x+h-3)(x-3)}}{h} Remember, multiplying by x3x3\frac{x-3}{x-3} is just like multiplying by one, so it doesn't change the *value* of the fraction, just its *form*.

STEP 6

Now, let's **expand** and **combine** those numerators: 2(x23x+hx3h)2x(x+h3)(x+h3)(x3)h=2x26x+2hx6h2x22hx+6x(x+h3)(x3)h \frac{\frac{2(x^2 -3x +hx -3h) - 2x(x+h-3)}{(x+h-3)(x-3)}}{h} = \frac{\frac{2x^2 -6x +2hx -6h - 2x^2 -2hx +6x}{(x+h-3)(x-3)}}{h} Notice how we carefully distributed the 22 and the 2x-2x.

STEP 7

Now, look at that!
A lot of terms **add to zero**!
We're left with: 6h(x+h3)(x3)h \frac{\frac{-6h}{(x+h-3)(x-3)}}{h}

STEP 8

Finally, we can **divide** both the numerator and denominator by hh.
Remember, dividing by hh is the same as multiplying by 1h\frac{1}{h}: 6h(x+h3)(x3)1h=6hh(x+h3)(x3) \frac{-6h}{(x+h-3)(x-3)} \cdot \frac{1}{h} = -\frac{6h}{h(x+h-3)(x-3)} Since hh=1\frac{h}{h} = 1, we get: 6(x+h3)(x3) -\frac{6}{(x+h-3)(x-3)}

STEP 9

The simplified difference quotient is 6(x+h3)(x3)-\frac{6}{(x+h-3)(x-3)}, which isn't in the list, so the answer is "Nothing in this list is correct."

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