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The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain - AQA - A-Level Maths: Pure - Question 10 - 2022 - Paper 3
Question 10
The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve $y = f(x)$ intersects the line $y = x$ at the p... show full transcript
Worked Solution & Example Answer:The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain - AQA - A-Level Maths: Pure - Question 10 - 2022 - Paper 3
Step 1
State the value of x which is not in the domain of f.
Answer
To determine the value of x that is not in the domain of f, we need to identify the points where the denominator equals zero. The denominator is given by:
Solving this equation:
Thus, the value of x which is not in the domain of f is .
Step 2
Show that P and Q are stationary points of the curve.
Answer
To show that P and Q are stationary points, we need to find the derivative of the function f(x) and set it to zero. First, we differentiate f(x):
Using the quotient rule:
Simplifying the derivative:
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Expand the numerator:
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Set the numerator to zero to find stationary points:
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Use the quadratic formula to solve for x:
This leads to two stationary points:
Thus, points P and Q are stationary points of the curve.
Step 3
Using set notation, state the range of f.
Answer
To find the range of f, we need to analyze the behavior of the function:
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As the domain of f excludes , we check the limits:
- As approaches from the left, .
- As approaches from the right, .
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Lastly, we check for horizontal asymptotes as or :
- As , (f(x) \to 1).
- As , (f(x) \to 1).
Thus, the range of f can be expressed in set notation as: