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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 1.3 New Functions from Old Functions

Subsection 1.3.1 Before Class

https://mymedia.ou.edu/media/1.3-1/1_xf5w60k6
Figure 4. Pre-Class Video 1

Subsubsection 1.3.1.1 Combinations of Functions

Functions can be combined in several ways to create new functions
Function Operations.
Let \(f\) be a function with domain \(A\) and \(g\) be a function with domain \(B\text{.}\)
  • Addition/Subtraction: \(h(x) = (f\pm g)(x) = f(x)\pm g(x)\text{.}\) The domain of \(h(x)\) is \(A\cap B\text{.}\)
  • Multiplication: \(j(x) = (fg)(x) = f(x)\cdot g(x)\text{.}\) The domain of \(j\) is \(A\cap B\text{.}\)
  • Division: \(k(x) = \lrpar{\dfrac{f}{g}}(x) = \dfrac{f(x)}{g(x)}\text{,}\) provided that \(g(x)\neq 0\text{.}\) The domain of \(k(x)\) is \(\lrbrace{x\in A\cap B\mid g(x)\neq 0}\)
Example 1.3.1.
If \(f(x) = \sqrt{3-x}\) and \(g(x) =\sqrt{x^2-1}\text{,}\) write (i) \(f+g\text{,}\) (ii) \(f-g\text{,}\) (iii) \(fg\text{,}\) and (iv) \(f/g\) and find their domains.
Solution.
  1. \((f+g)(x) = \sqrt{3-x}+\sqrt{x^2-1}\) with domain \((-\infty,-1]\)
  2. \((f-g)(x) = \sqrt{3-x}-\sqrt{x^2-1}\) with domain \((-\infty,-1]\)
  3. \((fg)(x) = \sqrt{-x^3+3x^2+x-3}\) with domain \((-\infty,-1]\)
  4. \((f/g)(x) = \sqrt{\dfrac{3-x}{x^2-1}}\) with domain \((-\infty,-1)\)
Example 1.3.2.
Find the domain of the function \(f(x) = 6x^4 + 14x + \dfrac{\cos x}{x^2-1}\)
Solution.
\((-\infty,-1)\cup(-1,1)\cup (1,\infty)\)
Example 1.3.3.
Find the domain of the function \(\dfrac{\sqrt{x}}{x^2-x-6}\text{.}\)
Solution.
\([0,3)\cup (3,\infty) \)
Example 1.3.4.
Find the domain of the function \(x|x|\text{,}\) and sketch the graph.
Solution.
The domain is \((-\infty,\infty)\text{.}\)
The graph of the function \(x|x|\) on the interval \([-3,3]\)

Subsubsection 1.3.1.2 Transformations of Functions

Vertical and Horizontal Shifts.
Let \(c \gt 0\text{.}\) To obtain the graph of
  • \(y = f(x) + c\text{,}\) shift the graph of \(y = f(x)\) \(c\) units up.
  • \(y = f(x) - c\text{,}\) shift the graph of \(y = f(x)\) \(c\) units down.
  • \(y = f(x+c)\text{,}\) shift the graph of \(y = f(x)\) \(c\) units left.
  • \(y = f(x-c)\text{,}\) shift the graph of \(y = f(x)\) \(c\) units right.
Vertical and Horizontal Stretching/Reflecting.
Let \(c \gt 1\text{.}\) To obtain the graph of
  • \(y = c\cdot f(x)\text{,}\) stretch the graph of \(y = f(x)\) vertically by a factor of \(c\text{.}\)
  • \(y = \lrpar{\dfrac{1}{c}}f(x)\text{,}\) shrink the graph of \(y = f(x)\) vertically by a factor of \(c\text{.}\)
  • \(y = f(c\cdot x)\text{,}\) shrink the graph of \(y = f(x)\) horizontally by a factor of \(c\text{.}\)
  • \(y = f\lrpar{\dfrac{x}{c}}\text{,}\) stretch the graph of \(y = f(x)\) horizontally by a factor of \(c\text{.}\)
  • \(y = -f(x)\text{,}\) reflect the graph of \(y = f(x)\) about the \(x-\)axis.
  • \(y = f(-x)\text{,}\) reflect the graph of \(y = f(x)\) about the \(y-\)axis.

Subsection 1.3.2 Pre-Class Activities

Example 1.3.5.

If \(f(x) = \sqrt{3-x}\) and \(g(x) = \sqrt{x^2-1}\text{,}\) write \((g/f)(x)\) and find its domain.
Solution.
\((f/g)(x) = \sqrt{\dfrac{x^2-1}{3-x}}\) and its domain is \((-\infty,-1]\cup [1,3)\cup (3,\infty)\)

Example 1.3.6.

Why do we even need to think about domain when working with a function?
Solution.
Answers vary

Example 1.3.7.

Describe the transformations needed to transform the graph \(f(x) = x^3\) into the graph of \(g(x) = -4(x+3)^3 + 1\text{.}\)
Solution.
  • Move left 3
  • Then reflect vertically
  • Then stretch vertically by a factor of 4
  • Then shift up 1

Subsection 1.3.3 In Class

Example 1.3.8.

The graph of \(y = \sqrt{x}\) is given to you below. Match the transformation with the appropriate transformed graph below. All graphs are of the form \(2f(x), f(2x), f\left(\dfrac{1}{2}x\right)\text{,}\) etc.
A graph of \(\sqrt{x}\text{,}\) from \(x=0\) to \(x=4\text{.}\) There is a labeled point at \((4,2)\text{.}\)
  1. Vertical shrink
  2. Vertical stretch
  3. Vertical shift
  4. Horizontal stretch
  5. Horizontal shrink
  6. Horizontal shift
  7. Vertical reflection
  1. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,4)\text{,}\) but the point at the origin has not moved.
  2. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,0)\text{,}\) and the point at the origin is now at \((0,-2)\text{.}\)
  3. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,\sqrt{8})\text{,}\) but the point at the origin has not moved.
  4. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,-2)\text{,}\) but the point at the origin has not moved.
  5. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,\sqrt{6})\text{,}\) and the point at the origin has moved to \((-2,0)\text{.}\)
  6. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,1)\text{,}\) but the point at the origin has not moved.
  7. The original graph of \(\sqrt{x}\) has been transformed so that the point at \((4,2)\) is now at \((4,\sqrt{2})\text{,}\) but the point at the origin has not moved.
Solution.
  1. Vertical stretch
  2. Vertical shift
  3. Horizontal shrink
  4. Vertical reflection
  5. Horizontal shift
  6. Vertical shrink
  7. Horizontal stretch

Example 1.3.9.

Sketch the graph of the function \(f(x)=x^2-2x+3\) by applying transformations to the base graph of \(f(x)=x^2\text{.}\)
A graph of \(f(x)=x^2\) on the interval \([-2,2]\text{.}\)
A blank graph on which to draw the transformed graph, \(f(x)=x^2-2x+3\text{.}\)
Solution.
The graph of \(f(x) = x^2-2x+3\) on \([-1,3]\)

Example 1.3.10.

Sketch the graphs of the given functions by using transformations to a base graph:
  1. \(\displaystyle f(x) = \dfrac{2}{x+3}\)
  2. \(\displaystyle g(x) = 1-\cos 2x\)
  3. \(\displaystyle h(x) = |x^2-1|\)
Solution.
  1. The graph of \(f(x) = \dfrac{-2}{x+3}\) on \([-8,8]\)
  2. The graph of \(g(x) = 1-\cos 2x\) on \([-2\pi,2\pi]\)
  3. The graph of \(g(x) = |x^2-1|\) on \([2,2]\)

Function Composition.

Given two function \(f\) and \(g\text{,}\) the composite function \(f\circ g\) is defined by \((f\circ g)(x) = f(g(x))\text{.}\)
The domain of \(f\circ g\) is the set of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\text{.}\)

Example 1.3.11.

For each function below, (1) find the domain of the composite function, (2) completely decompose the function into smaller ones.
  1. \(\displaystyle f(x) = \dfrac{1}{x+2}\)
  2. \(\displaystyle q(x) = (2x+1)^5\)
  3. \(\displaystyle s(h) = \sin \left(5h^2 + \dfrac{1}{h}\right)\)
  4. \(\displaystyle y(r) = \dfrac{5.317}{(2r^5 + 1.7)^2}\)
  5. \(\displaystyle g(x) = \tan (x^2)\)
Solution.
  1. The domain is \((-\infty,-2)\cup (-2,\infty)\text{.}\) The function can decompose as \(f(x) = a(b(x))\text{,}\) where \(a(x) = \dfrac{1}{x}, b(x) = x+2\)
  2. The domain is \((-\infty,\infty)\text{.}\) The function can decompose as \(q(x) = a(b(x))\text{,}\) where \(a(x) = x^5, b(x) = 2x+1\)
  3. The domain is \((-\infty,0)\cup (0,\infty)\text{.}\) The function can decompose as \(s(h)) = a(b(h))\text{,}\) where \(a(h) = \sin h, b(h) = 5h^2 + \dfrac{1}{h}\)
  4. The domain is \(\lrpar{-\infty, \sqrt[5]{\dfrac{-1.7}{2}}}\cup \lrpar{\sqrt[5]{\dfrac{-1.7}{2}},\infty}\text{.}\) The function can decompose as \(y(r) = a(b(r))\text{,}\) where \(a(r) = \dfrac{5.317}{r^2}, b(x) = 2r^5+1.7\)
  5. The domain is \(x\neq \sqrt{\dfrac{\pi}{2}+\pi k}\text{,}\) where \(k\in\Z\text{.}\) The function can decompose as \(g(x) = a(b(x))\text{,}\) where \(a(x) = \tan x, b(x) = x^2\)

Example 1.3.12.

If \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{2-x}\text{,}\) find
  1. \(\displaystyle f\circ g\)
  2. \(\displaystyle g\circ f\)
  3. \(\displaystyle f\circ f\)
  4. \(\displaystyle g\circ g\)
Solution.
  1. \(\displaystyle \sqrt{\sqrt{2-x}} =\sqrt[4]{2-x}\)
  2. \(\displaystyle \sqrt{2-\sqrt{x}}\)
  3. \(\displaystyle \sqrt{\sqrt{x}} = \sqrt[4]{x}\)
  4. \(\displaystyle \sqrt{2-\sqrt{2-x}}\)

Example 1.3.13.

Let \(k(x) = \sec(x^2)\tan(x^2)\text{.}\) Find \(f,g\) such that \(k(x) = f(g(x))\text{.}\)
Solution.
\(f(x) = \sec(x)\tan(x),g(x) = x^2\)

Example 1.3.14.

Let \(f(x) = \cos^2 (x^2 + 9)\text{.}\) Find functions \(a,b,c\) such that \(f(x) = (a\circ b\circ c)(x)\)
Solution.
\(a(x) = x^2, b(x) = \cos(x), c(x) = x^2+9\)

Example 1.3.15.

Given \(f(x) = x^2 + x -1\) and \(g(x) = 2-x\text{,}\) what is the equation of \(y =(f\circ g)(x)\text{?}\)
Solution.
\(y = 5-5x+x^2\)

Example 1.3.16.

If \(g(1) = 3\text{,}\) then what point must be on the graph of \(h(t) = -2g(t-1) + 6\text{?}\)
Solution.
\((2,0)\)

Subsection 1.3.4 After Class Activities

Example 1.3.17.

For each function below, (i) identify the base graph, (ii) identify any transformation applied to the base graph, and (iii) sketch a rough graph of the function.
  1. \(\displaystyle y = 2\cos 3x\)
  2. \(\displaystyle y = 3\sqrt{x+1}\)
  3. \(\displaystyle y = |\sqrt{x} - 1|\)
Solution.
  1. The base graph is \(\cos x\text{.}\) We transform by compressing horizontally by 3, then stretching vertically by 2.
    This is the graph of \(y=2\cos(3x)\) on the interval \([0,2\pi]\)
  2. The base graph is \(\sqrt{x}\text{.}\) First we shift left by 1, then stretch vertically by 3.
    This is the graph of \(3\sqrt{x+1}\) on the interval \([0,4]\)
  3. This is the graph of \(|\sqrt{x}-1|\) on the interval \([0,4]\)

Example 1.3.18.

Below is a table of input/output values for functions \(f\) and \(g\text{.}\)
\(x\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\)
\(f(x)\) \(3\) \(1\) \(4\) \(2\) \(2\) \(5\)
\(g(x)\) \(6\) \(3\) \(2\) \(1\) \(2\) \(3\)
Evaluate each expression below:
  1. \(\displaystyle f(g(1))\)
  2. \(\displaystyle g(f(1))\)
  3. \(\displaystyle f(f(1))\)
  4. \(\displaystyle g(g(1))\)
  5. \(\displaystyle (g\circ f)(3)\)
  6. \(\displaystyle (f\circ g)(6)\)
Solution.
  1. 5
  2. 2
  3. 4
  4. 3
  5. 1
  6. 4

Example 1.3.19.

If you invest \(x\) dollars at at 6% interest compounded annually, then the amount \(A(x)\) of the investment after one year is \(A(x) = 1.06x\text{.}\)
  1. Find \(A\circ A\circ A\circ A\) and \(A\circ A\circ A\circ A\circ A\text{.}\)
  2. What do the compositions represent in real-world terms?
  3. Generalize your work in part (a); find a formula for the composition of \(n\) copies of \(A\text{.}\)
Solution.
  1. \(A\circ A\circ A\circ A = (1.06)^4x\) and \(A\circ A\circ A\circ A\circ A = (1.06)^5x\)
  2. The compositions represent compounds of the investment.
  3. \(\displaystyle (1.06)^nx\)

Example 1.3.20.

If \(g(x) = 2x + 1\) and \(h(x) =4x^2 + 4x + 7\text{,}\) find a function \(f\) such that \(f\circ g = h\text{.}\)
Solution.
\(f(x) = x^2+6\)

Subsection 1.3.5 Section 1.3 Resources

Subsubsection 1.3.5.1 Function Operations

Subsubsection 1.3.5.2 Graph Transformations

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