Logarithmic and Exponential Functions Structured Questions | Cambridge O Level Additional Mathematics

Question 1 of 3 · 10 marks

Logarithmic and Exponential Functions · 1 - Integrated Exponential & Logarithmic Graphs, Inverses & Reflections

3 parts

Additional Mathematics 4037 · Question 1Paper 2 · Calculator[10 marks]

The function $f$ is defined by $f(x) = e^{2x} - 4$ for $x \in \mathbb{R}$ .

(a)

Find an expression for $f^{-1}(x)$ and state its domain and range.

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(b)

State the equation of the horizontal asymptote of $y = f(x)$ and the vertical asymptote of $y = f^{-1}(x)$ . Find the exact coordinates of the $x$ -intercept and $y$ -intercept of $y = f(x)$ .

[3]

(c)

Explain why the curves $y = f(x)$ and $y = f^{-1}(x)$ intersect on the line $y = x$ . Show that the $x$ -coordinate of any point of intersection satisfies $e^{2x} - x - 4 = 0$ , and verify by substitution that $x = 0$ is NOT a solution.

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