A particle’s position along the x-axis at time $t$ is governed by the function $x(t)$. Its velocity, $v(t)$, is given by $v(t) = rac{dx}{dt}$. Two researchers, Alice and Bob, propose different models relating the particle’s position to its velocity. Alice suggests that $v(t) = f(x(t))$, while Bob believes $x(t) = g(v(t))$, where $f(x)$ and $g(v)$ are differentiable functions. Assume both models are valid for the particle’s motion.
c. Given that $f(x) = x^2 + 1$ and $g(v) = \sqrt{v-1}$, evaluate whether the composition $f(g(v))$ or $g(f(x))$ is mathematically valid for all real numbers. Justify your answer with reference to the domains and ranges of $f(x)$ and $g(v)$.
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 4 marks, testing your understanding of Composition of Functions. It falls under Algebra, number and structure in Unit 3: Mathematical Methods Unit 3. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Extend introductory study of functions, algebra, calculus. Focus on functions, relations, graphs, algebra, and applications of derivatives.
Covers algebra of functions, inverse functions, and solutions of equations and systems of equations.
composition of functions, where $f$ composite $g, f \circ g$, is defined by $(f \circ g)(x)=f(g(x))$ given $r_{g} \subseteq d_{f}$
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