Multi-Function Intersection Analysis

00:00 10 marks

Analyze the intersections of the following functions:

1. A linear function \(f(x) = 2x + 1\)
2. A quadratic function \(g(x) = x^2 – 4\)
3. An exponential function \(h(x) = 3e^{x}\)

Proceed as follows:

Part a

4 minutes 3 marks

Question

Set up the equations to find the intersections between each pair of functions:

(i) \(f(x) = g(x)\),
(ii) \(f(x) = h(x)\), and
(iii) \(g(x) = h(x)\).

Provide the equations in simplified form.

Hint Hint

Hint

Equate the expressions for each pair of functions and simplify the resulting equations where possible.

Answer Answer
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Complete Answer

Equations:
(i) \(2x + 1 = x^2 – 4\) simplifies to \(x^2 – 2x – 5 = 0\),
(ii) \(2x + 1 = 3e^{x}\),
(iii) \(x^2 – 4 = 3e^{x}\).

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Part b

5 minutes 4 marks

Question

Using a graphical calculator, estimate the approximate values of the x-coordinates of the intersection points obtained from the equations in part (a). List all x-coordinates to three significant figures.

Hint Hint

Hint

Use the intersection function on your calculator to find approximate roots. Graph each pair of functions to visually confirm the intersections and ensure you have found all of them within a reasonable window.

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Approximately, for \(f(x)=g(x)\): \(x \approx -1.45\) and \(x \approx 3.45\); for \(f(x)=h(x)\): \(x \approx 0.251\); for \(g(x)=h(x)\): No real solution.

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Part c

3 minutes 3 marks

Question

Sketch the graphs of the three functions on the same axes and discuss how the number and nature of intersection points reflect differences in their behavior. Mention any discrepancies between analytical and calculator-based approaches.

Hint Hint

Hint

Consider the slopes, curvature, and asymptotic behavior of the functions and how these graphical features lead to the observed number of intersections. Think about the long-term behavior of each function as x approaches positive and negative infinity.

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The graph shows that the linear function and quadratic function intersect at two points, reflecting the line crossing the parabola twice. The exponential function intersects the linear function at one point, indicating the exponential curve crossing the line once. The exponential function does not intersect the quadratic function, showing that the exponential curve grows faster than the parabola and eventually stays above it. Calculator approximations confirm these observations, with minor discrepancies arising from rounding to three significant figures.

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