Preparing for Calculus

P.3 Assess Your Understanding

Concepts and Vocabulary

Question

If the domain of a function \(f\) is \(\{ x|0\leq x\leq 7\} \) and the domain of a function \(g\) is \(\{ x|{-}2\leq x\leq 5\} ,\) then the domain of the sum function \(f+g\) is _____.

Question

True or False \(\quad\) If \(f\) and \(g\) are functions, then the domain of \(\dfrac{f}{g}\) consists of all numbers \(x\) that are in the domains of both \(f\) and \(g\).

Question

True or False \(\quad\) The domain of \( f\cdot g \) consists of the numbers \(x\) that are in the domains of both \(f\) and \(g\).

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True or False \(\quad\) The domain of the composite function \( f\circ g\) is the same as the domain of \(g( x) \).

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True or False \(\quad\) The graph of \(y=-f( x)\) is the reflection of the graph of \(y=f( x) \) about the \(x\)-axis.

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True or False \(\quad\) To obtain the graph of \(y=f(x+2) -3\), shift the graph of \(y=f( x) \) horizontally to the right 2 units and vertically down 3 units.

Question

True or False \(\quad\) Suppose the \(x\)-intercepts of the graph of the function \(f\) are \(-3\) and \(2\). Then the \(x\)-intercepts of the graph of the function \(y=2f( x) \) are \(-3\) and \(2.\)

Question

Suppose that the graph of a function \(f\) is known. Then the graph of the function \(y=f( x-2) \) can be obtained by a(n)_____ shift of the graph of \(f\) to the _____ a distance of 2 units.

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Suppose that the graph of a function \(f\) is known. Then the graph of the function \(y=f( -x) \) can be obtained by a reflection about the _____-axis of the graph of \(f\).

Question

Suppose the \(x\)-intercepts of the graph of the function \(f\) are \(-2\), \(1\), and \(5\). The \(x\)-intercepts of \(y=f( x+3) \) are _____, _____, and _____.

Practice Problems

In Problems 11–14, the functions \(f\) and \(g\) are given. Find each of the following functions and determine their domain:

\(( f+g) ( x) \)
\((f-g) ( x) \)
\(( f\cdot g) ( x) \)
\(\left( \dfrac{f}{g}\right) ( x)\)

Question

\(f( x) =3x+4\) and \(g( x) =2x-3\)

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\(f( x) =1+\dfrac{1}{x}\) and \(g( x) =\dfrac{1}{x}\)

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\(f( x) =\sqrt{x+1}\) and \(g( x) =\dfrac{2}{x}\)

Question

\(f( x) =\vert x\vert \) and \(g( x) =x\)

In Problems 15 and 16, for each of the functions \(f\) and \(g\), find:

\(( f\circ g) ( 4)\)
\(( g\circ f) ( 2)\)
\(( f\circ f) ( 1)\)
\(( g\circ g) ( 0)\)

Question

\(f( x) =2x\) and \(g( x) =3x^{2}+1\)

Question

\(f( x) =\dfrac{3}{x+1}\) and \(g( x) =\sqrt{x}\)

In Problems 17 and 18, evaluate each expression using the values given in the table.

Question

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f( x) \)	\(-7\)	\(-5\)	\(-3\)	\(-1\)	\(3\)	\(5\)	\(7\)
\(g( x)\)	\(8\)	\(3\)	\(0\)	\(-1\)	\(0\)	\(3\)	\(8\)

\(( f\circ g) ( 1) \)
\(( f\circ g) ( -1) \)
\(( g\circ f) ( -1) \)
\(( g\circ f) ( 1) \)
\(( g\circ g) ( -2) \)
\(( f\circ f) ( -1) \)

Question

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f( x)\)	\(11\)	\(9\)	\(7\)	\(5\)	\(3\)	\(1\)	\(-1\)
\(g( x)\)	\(-8\)	\(-3\)	\(0\)	\(1\)	\(0\)	\(-3\)	\(-8\)

\(( f\circ g) ( 1) \)
\(( f\circ g) ( 2) \)
\(( g\circ f) ( 2) \)
\(( g\circ f) ( 3) \)
\(( g\circ g) ( 1) \)
\(( f\circ f) ( 3) \)

In Problems 19 and 20, evaluate each composite function using the graphs of \(y=f( x) \) and \(y=g( x) \) shown in the figure below.

Question

\(( g\circ f) ( -1) \)
\(( g\circ f) ( 6) \)
\(( f\circ g) ( 6) \)
\(( f\circ g) ( 4) \)

Question

\(( g\circ f) ( 1) \)
\(( g\circ f) ( 5) \)
\(( f\circ g) ( 7) \)
\(( f\circ g) ( 2) \)

In Problems 21–26, for the given functions \(f\) and \(g\), find:

\(f\circ g\)
\(g\circ f\)
\(f\circ f\)
\(g\circ g\)

State the domain of each composite function.

Question

\(f( x) =3x+1\) and \(g( x) =8x\)

Question

\(f( x) =-x\) and \(g( x) =2x-4\)

Question

\(f( x) =x^{2}+1\) and \(g( x) =\sqrt{x-1}\)

Question

\(f( x) =2x+3\) and \(g( x) =\sqrt{x}\)

Question

\(f( x) =\dfrac{x}{x-1}\) and \(g( x) = \dfrac{2}{x}\)

Question

\(f( x) =\dfrac{1}{x+3}\) and \(g( x) =-\dfrac{2}{x}\)

In Problems 27–32, find functions \(f\) and \(g\) so that \(f\circ g=F\).

Question

\(F( x) = ( 2x+3) ^{4}\)

Question

\(F( x) = ( 1+x^{2}) ^{3}\)

Question

\(F( x) =\sqrt{x^{2}+1}\)

Question

\(F( x) =\sqrt{1-x^{2}}\)

Question

\(F( x) = \vert 2x+1\vert \)

Question

\(F( x) = \vert 2x^{2}+3\vert\)

In Problems 33–46, graph each function using the graphing techniques of shifting, compressing, stretching, and/or reflecting. Begin with the graph of a basic function and show all stages.

Question

\(f( x) =x^{3}+2\)

Question

\(g( x) =x^{3}-1\)

Question

\(h( x) =\sqrt{x-2}\)

Question

\(f( x) =\sqrt{x+1}\)

Question

\(g( x) =4\sqrt{x}\)

Question

\(f( x) =\dfrac{1}{2}\sqrt{x}\)

Question

\(f( x) =( x-1) ^{3}+2\)

Question

\(g( x) =3( x-2) ^{2}+1\)

Question

\(h( x) =\dfrac{1}{2x}\)

Question

\(f( x) =\dfrac{4}{x}+2\)

Question

\(G( x) =2 \vert 1-x\vert \)

Question

\(g( x) =-( x+1) ^{3}-1\)

Question

\(g( x) =-4\sqrt{x-1}\)

Question

\(f( x) =4\sqrt{2-x}\)

In Problems 47 and 48, the graph of a function \(f\) is illustrated. Use the graph of \(f\) as the first step in graphing each of the following functions:

\(F( x) =f( x) +3\)
\(G( x) =f( x+2) \)
\(P( x) =-f( x) \)
\(H( x) =f( x+1) -2\)
\(Q( x) =\dfrac{1}{2}f( x) \)
\(g( x) =f( -x) \)
\(h( x) =f( 2x)\)

Question

Period of a Pendulum \(\quad\) The period \(T\) (in seconds) of a simple pendulum is a function of its length \(l\) (in meters) defined by the equation \[ T=2\pi \sqrt{\dfrac{l}{g}} \] where \(g\approx 9.8\) meters / second\(^{2}\) is the acceleration due to gravity.

Use a graphing utility to graph the function \(T=T ( l).\)
Now graph the functions \(T=T ( l+1) ,\) \(T=T ( l+2) ,\) and \(T=T ( l+3) \).
Discuss how adding to the length \(l\) changes the period \(T\).
Now graph the functions \(T=T ( 2l) ,\) \(T=T ( 3l) ,\) and \(T=T ( 4l) \).
Discuss how multiplying the length \(l\) by 2, 3, and 4 changes the period \(T\).

Question

Suppose \(( 1,3) \) is a point on the graph of \(y=g( x) \).

What point is on the graph of \(y=g( x+3) -5?\)
What point is on the graph of \(y=-2g( x-2) +1?\)
What point is on the graph of \(y=g( 2x+3) ?\)