91Ó°ÊÓ

If \(f(x)=2 x+8\) and \(g(x)=3 x-2\), determine each of the following. a) \(f(g(1))\) b) \(f(g(-2))\) \(g(f(-4))\) d) \(g(f(1))\)

If \(f(x)=3 x+4\) and \(g(x)=x^{2}-1\), determine each of the following. a) \(f(g(a))\) b) \(g(f(a))\) c) \(f(g(x))\) d) \(g(f(x))\) e) \(f(f(x))\) f) \(g(g(x))\)

Let \(j(x)=x^{2}\) and \(k(x)=x^{3} .\) Does \(k(j(x))=j(k(x))\) for all values of \(x ?\) Explain.

A Ferris wheel rotates such that the angle, \(\theta,\) of rotation is given by \(\theta=\frac{\pi t}{15},\) where \(t\) is the time, in seconds. A rider's height, \(h,\) in metres, above the ground can be modelled by \(h(\theta)=20 \sin \theta+22\) a) Write the equation of the rider's height in terms of time. b) Graph \(h(\theta)\) and \(h(t)\) on separate sets of axes. Compare the periods of the graphs.

An alternating current-direct current (AC-DC) voltage signal is made up of the following two components, each measured in volts \((\mathrm{V}): V_{\mathrm{AC}}(t)=10 \sin t\) and \(V_{\mathrm{DC}}(t)=15\). a) Sketch the graphs of these two functions on the same set of axes. Work in radians. b) Graph the combined function \(V_{A C}(t)+V_{\mathrm{DC}}(t)\) c) Identify the domain and range of \(V_{\mathrm{AC}}(t)+V_{\mathrm{DC}}(t)\) d) Use the range of the combined function to determine the following values of this voltage signal. i) minimum ii) maximum

According to Einstein's special theory of relativity, the mass, \(m,\) of a particle moving at velocity \(v\) is given by \(m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\), where \(m_{0}\) is the particle's mass at rest and \(c\) is the velocity of light. Suppose that velocity, \(v,\) in miles per hour, is given as \(V=t^{3}\). a) Express the mass as a function of time. b) Determine the particle's mass at time \(t=\sqrt[3]{\frac{c}{2}}\) hours.

An even function satisfies the property \(f(-x)=f(x)\) for all \(x\) in the domain of \(f(x)\) An odd function satisfies the property \(f(-x)=-f(x)\) for all \(x\) in the domain of \(f(x)\). Devise and test an algebraic method to determine if the sum of two functions is even, odd, or neither. Show by example how your method works. Use at least three of the functions you have studied: absolute value, radical, polynomial, trigonometric, exponential, logarithmic, and rational.