Chapter 5

According to Einstein's Theory of Special Relativity, the observed mass \(m\) of an object is a function of how fast the object is traveling. Specifically, $$ m(x)=\frac{m_{r}}{\sqrt{1-\frac{x^{2}}{c^{2}}}} $$ where \(m(0)=m_{r}\) is the mass of the object at rest, \(x\) is the speed of the object and \(c\) is the speed of light. (a) Find the applied domain of the function. (b) Compute \(m(.1 c), m(.5 c), m(.9 c)\) and \(m(.999 c)\). (c) As \(x \rightarrow c^{-}\), what happens to \(m(x) ?\) (d) How slowly must the object be traveling so that the observed mass is no greater than 100 times its mass at rest?

Let \(g(x)=-x, h(x)=x+2, j(x)=3 x\) and \(k(x)=x-4\). In what order must these functions be composed with \(f(x)=\sqrt{x}\) to create \(F(x)=3 \sqrt{-x+2}-4 ?\)

Find the inverse of \(k(x)=\frac{2 x}{\sqrt{x^{2}-1}}\).

What linear functions could be used to transform \(f(x)=x^{3}\) into \(F(x)=-\frac{1}{2}(2 x-7)^{3}+1 ?\) What is the proper order of composition?

Suppose Fritzy the Fox, positioned at a point \((x, y)\) in the first quadrant, spots Chewbacca the Bunny at (0,0) . Chewbacca begins to run along a fence (the positive \(y\) -axis) towards his warren. Fritzy, of course, takes chase and constantly adjusts his direction so that he is always running directly at Chewbacca. If Chewbacca's speed is \(v_{1}\) and Fritzy's speed is \(v_{2},\) the path Fritzy will take to intercept Chewbacca, provided \(v_{2}\) is directly proportional to, but not equal to, \(v_{1}\) is modeled by $$ y=\frac{1}{2}\left(\frac{x^{1+v_{1} / v_{2}}}{1+v_{1} / v_{2}}-\frac{x^{1-v_{1} / v_{2}}}{1-v_{1} / v_{2}}\right)+\frac{v_{1} v_{2}}{v_{2}^{2}-v_{1}^{2}} $$ (a) Determine the path that Fritzy will take if he runs exactly twice as fast as Chewbacca; that is, \(v_{2}=2 v_{1} .\) Use your calculator to graph this path for \(x \geq 0 .\) What is the significance of the \(y\) -intercept of the graph? (b) Determine the path Fritzy will take if Chewbacca runs exactly twice as fast as he does; that is, \(v_{1}=2 v_{2}\). Use your calculator to graph this path for \(x>0 .\) Describe the behavior of \(y\) as \(x \rightarrow 0^{+}\) and interpret this physically. (c) With the help of your classmates, generalize parts (a) and (b) to two cases: \(v_{2}>v_{1}\) and \(v_{2}

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(f \circ g)(3)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$f(g(-1))$$

Show that \(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}}=x\) for all \(x \geq 0\).

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(f \circ f)(0)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(f \circ g)(-3)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(g \circ f)(3)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$g(f(-3))$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(g \circ g)(-2)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$(g \circ f)(-2)$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$g(f(g(0)))$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$f(f(f(-1)))$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$f(f(f(f(f(1)))))$$

Let \(f\) be the function defined by $$f=\\{(-3,4),(-2,2),(-1,0),(0,1),(1,3),(2,4),(3,-1)\\}$$ and let \(g\) be the function defined $$g=\\{(-3,-2),(-2,0),(-1,-4),(0,0),(1,-3),(2,1),(3,2)\\}$$ Find the value if it exists. $$\underbrace{(g \circ g \circ \cdots \circ g)}_{n \text { times }}(0)$$

The volume \(V\) of a cube is a function of its side length \(x\). Let's assume that \(x=t+1\) is also a function of time \(t,\) where \(x\) is measured in inches and \(t\) is measured in minutes. Find a formula for \(V\) as a function of \(t\).

Suppose a local vendor charges \(\$ 2\) per hot dog and that the number of hot dogs sold per hour \(x\) is given by \(x(t)=-4 t^{2}+20 t+92,\) where \(t\) is the number of hours since 10 AM, \(0 \leq t \leq 4\). (a) Find an expression for the revenue per hour \(R\) as a function of \(x .\) (b) Find and simplify \((R \circ x)(t) .\) What does this represent? (c) What is the revenue per hour at noon?

Discuss with your classmates how 'real-world' processes such as filling out federal income tax forms or computing your final course grade could be viewed as a use of function composition. Find a process for which composition with itself (iteration) makes sense.