Chapter 3

Two brothers, Max and Eli, are experimenting with their walkie-talkies. (A walkie-talkie is a combined radio transmitter and receiver light enough to allow the user to walk and talk at the same time.) The quality of the transmission, \(Q\), is a function of the distance between the two walkie- talkies. We will model it as being inversely proportional to this distance. At time \(t=0\) Max is 100 feet north of Eli. Max walks north at a speed of 300 feet per minute while Eli walks east at a speed of 250 feet per minute. All the time they are talking on their walkie-talkies. (a) Write a function \(f\) such that \(Q=f(d)\), where \(d\) is the distance between the brothers. Your function will involve an unknown constant. (b) Write a function \(g\) that gives the distance between the brothers at time \(t\). (c) Find \(f(g(t))\). What does this composite function take as input and what does it give as output?

\(A\) and \(B\) are points on the graph of \(k(x)\). The \(x\) -coordinate of point \(A\) is 6 and the \(x\) -coordinate of point \(B\) is \((6+h)\). Write mathematical expressions, using functional notation, for each of the following. (a) The change in value of the function from point \(A\) to point \(B\) (b) The average rate of change of the function \(k\) over the interval \([6,6+h]\) (c) Suppose that the average rate of change of the function \(k\) over the interval \([6,6+h]\) is \(-5\). The functions \(f, g\), and \(h\) are de ned as follows: $$ f(x)=k(x)+2, \quad g(x)=k(x+2), \quad h(x)=2 k(x). $$ i. Which of the following must also be equal to \(-5\) ? A. The average rate of change of the function \(f\) over the interval \([6,6+h]\) B. The average rate of change of the function \(g\) over the interval \([6,6+h]\) C. The average rate of change of the function \(h\) over the interval \([6,6+h]\) ii. One of the functions \(f, g\), and \(h\) has an average value of \(-10\) on the interval \([6,6+h]\). Which is it? Explain brie y.

If \(h(x)=f(g(x))\), then \(x\) is in the domain of \(h\) if and only if \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f .\) In other words, \(x\) must be a valid input for \(g\) and \(g(x)\) must be a valid input for \(f\). (a) If \(h(x)=f(g(x))\), where \(g(x)=\sqrt{x}\) and \(f(x)=x^{2}\), what is the largest possible domain of \(h\) ? For all \(x\) in its domain, \(h(x)=x\). Why is the domain not \((-\infty, \infty)\) ? (b) If \(h(x)=f(g(x))\), where \(g(x)=\frac{1}{x-1}\) and \(f(x)=\frac{1}{x+3}\), what is the largest possible domain of \(h ?\) (There are two numbers that must be excluded from the domain.)

Let \(f(x)=\frac{2 x}{x+3}\) and \(g(x)=\frac{1}{x+1}\). (a) Find \(f(g(2))\). (b) Find \(f(g(x))\) and simplify your answer. Be sure that your answer is in agreement with the concrete case from part (a).

Let \(f(x)=\frac{x}{x+3}\) and \(g(x)=\frac{3 x}{1-x}\). (a) Find \(f(g(2))\) and \(g(f(2))\). (b) Find \(f(g(x))\) and \(g(f(x))\). (c) What does part (b) suggest about the relationship between \(f\) and \(g\) ?

If the function \(m(t)=\frac{1}{t+2}\) and \(h(t)=t-2\), then is it ever true that \(m(h(t))=h(m(t))\) ?

The functions \(R(x), K(x), D(x)\), and \(L(x)\) are de ned as follows: $$R(x)=\frac{1}{x^{2}}, \quad K(x)=|x|, \quad D(x)=x+3, \quad L(x)=-5 x .$$ Evaluate the following expressions. (Be sure to give simpli ed expressions whenever possible.) (a) \(R(K(L(x)))\) (b) \(R(L(R(x)))\) (c) \(R(K(x))\) (d) \(R(D(R(x)))\)

Let \(f(x)=|x|, g(x)=\sqrt{x}\), and \(h(x)=x-2\). Find the domain for each of the following. (a) \(j(x)=g(h(x))\) (b) \(k(x)=h(g(x))\)

Let \(f(x)=|x|, g(x)=\sqrt{x}\), and \(h(x)=x-2\). Find the domain for each of the following. (a) \(l(x)=g(f(x))\) (b) \(m(x)=g(h(f(x)))\)

Let \(f(x)=|x|, g(x)=\sqrt{x}\), and \(h(x)=x-2\). Find the domain for each of the following. (a) \(p(x)=h(g(h(x)))\) (b) \(q(x)=f(h(g(x)))\)

Let \(f(x)=x^{2}+9, g(x)=\sqrt{x}\), and \(h(x)=g(f(x))\). Find the average rate of change of \(h\) over the following intervals. (a) \([-4,4]\) (b) \([0,4]\) (c) \([4,4+k]\)

Find \(h(x)=f(g(x))\) and \(j(x)=g(f(x)) .\) What are the domains of \(h\) and \(j\) ? $$ f(x)=x^{2}+9 \text { and } g(x)=\frac{1}{\sqrt{x}} $$

Find \(h(x)=f(g(x))\) and \(j(x)=g(f(x)) .\) What are the domains of \(h\) and \(j\) ? $$ f(x)=\frac{2}{x+2} \text { and } g(x)=x-2 $$

Find \(h(x)=f(g(x))\) and \(j(x)=g(f(x)) .\) What are the domains of \(h\) and \(j\) ? $$ f(x)=x^{2} \text { and } g(x)=-2 x+3 $$

Find \(h(x)=f(g(x))\) and \(j(x)=g(f(x)) .\) What are the domains of \(h\) and \(j\) ? $$ f(x)=\frac{x}{x-3} \text { and } g(x)=\frac{2}{x} $$

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=a x+b \text { and } g(x)=c x+d $$

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=3 x+2 \text { and } g(x)=5 x-1 $$

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=2 x+3 \text { and } g(x)=x^{2}-1 $$

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=\frac{3}{x+1} \text { and } g(x)=\frac{2 x}{x-5} $$

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=\sqrt{x} \text { and } g(x)=\sqrt{x-3} $$

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(g(f(2))\) (b) \(f(g(2))\)

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(g\left(f\left(\frac{1}{3}\right)\right)\) (b) \(f\left(g\left(\frac{1}{3}\right)\right)\)

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(g(f(1))\) (b) \(f(g(1))\)

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(f(f(2))\) (b) \(g(g(-1))\)

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \(f(g(x))\) (b) \(g(f(x))\)

Let \(f(x)=\frac{1}{x}+x\) and \(g(x)=\frac{2 x}{x^{2}+1} .\) Evaluate the following expressions. (a) \((f \circ f)(x)\) (b) \((f \circ f \circ f)(x)\)

(a) Suppose \(f\) and \(g\) are both even functions. What can be said about \((f+g)(x)\) ? \((f g)(x) ?\) (b) Suppose \(f\) and \(g\) are both odd functions. What can be said about \((f+g)(x)\) ? \((f g)(x) ?\) (c) Suppose \(f\) is an even function and \(g\) is an odd function. What can be said about \((f+g)(x) ?(f g)(x) ?\)

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ h(f(x))+h(g(x)) $$

Let \(f(x)=2 x^{2}, g(x)=x+1\), and $h(x)=\frac{1}{x} . If what is written is an expression, simplify it. If it is an equation, solve it. $$ g(x) h(f(x))+f(x) h(g(x)) $$

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ g(x) h(f(x))=1 $$

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ 2 h(f(x) g(x))+h(3 g(x)) $$

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ h(f(x)+3 g(x))=h(2) $$

Let \(f(x) = 2 x^{2}, g(x) = x+1\), and \(h(x) = \frac{1}{x}\) . If what is written is an expression, simplify it. If it is an equation, solve it. $$ f(g(x))=10 $$