A function $f$ is defined by
$$f(x) = \frac{-x}{\sqrt{2x - 2}}$$
6 (a) State the maximum possible domain of $f$ - AQA - A-Level Maths Pure - Question 6 - 2018 - Paper 3
Question 6
A function $f$ is defined by
$$f(x) = \frac{-x}{\sqrt{2x - 2}}$$
6 (a) State the maximum possible domain of $f$.
6 (b) Use the quotient rule to show that
$$f'(x... show full transcript
Worked Solution & Example Answer:A function $f$ is defined by
$$f(x) = \frac{-x}{\sqrt{2x - 2}}$$
6 (a) State the maximum possible domain of $f$ - AQA - A-Level Maths Pure - Question 6 - 2018 - Paper 3
Step 1
State the maximum possible domain of $f$
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Answer
To determine the maximum possible domain of the function f(x)=2x−2−x, we must ensure that the denominator is not zero and that the expression under the square root is non-negative.
The expression 2x−2≥0 implies that
x≥1.
Additionally, the term sqrt2x−2 must not equal zero, therefore:
2x−2>0⇒x>1.
Thus, the maximum possible domain of f is x>1, or in interval notation:
[ (1, \infty) ]
Step 2
Use the quotient rule to show that
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Answer
To find the derivative f′(x) using the quotient rule, where
f(x)=h(x)g(x),
where g(x)=−x and h(x)=2x−2.
The quotient rule states that:
f′(x)=(h(x))2g′(x)h(x)−g(x)h′(x).
First, calculate g′(x) and h′(x):
g′(x)=−1
For h(x)=(2x−2)1/2, using the chain rule,
h′(x)=21(2x−2)−1/2⋅2=2x−21.
Substitute these into the quotient rule:
f′(x)=(2x−2)2(−1)(2x−2)−(−x)(2x−21)
