Skip to main content

MAT 151: College Algebra with Trigonometry

Section 1.1 Families of Functions

Goals:
🔗
  • F: Be able to identify and sketch graphs of basic functions, and identify important features of these functions.
    🔗
    🔗
🔗
Families of functions are groups of functions that have common graphical and algebraic characteristics. Here, we will explore several families of functions and determine which characteristics are common to all functions in a given family.
🔗

Definition 1.1.1.

A function is even if and only if \(f(-x)=f(x)\) for all \(x\) in the domain of \(f\text{.}\) A function is odd if and only if \(f(-x)=-f(x)\) for all \(x\) in the domain of \(f\text{.}\)
🔗
🔗

Problem 1.1.2.

For each function, decide if the function is even, odd or neither even nor odd.
🔗
  1. The graph of \(f(x)\) shown in Figure 1.1.3 below.
    The graph of \(f(x)\) is a piecewise constant (step) function with a jump at \(x = 0\text{.}\) For \(x > 0\text{,}\) the graph is a horizontal line at \(y = 1\) with an open circle at \((0, 1)\text{.}\) For \(x < 0\text{,}\) the graph is a horizontal line at \(y = -1\) with an open circle at \((0, -1)\text{.}\) At \(x = 0\text{,}\) there is a filled point at \((0, 0)\text{,}\) indicating \(f(0) = 0\text{.}\) The function is discontinuous at \(x = 0\text{.}\)
    Figure 1.1.3. Graph of \(f(x)\text{.}\)
    🔗
    🔗
    🔗
  2. The graph of \(g(x)\) shown in Figure 1.1.4 below.
    The graph of \(g(x)\) is an upward-opening quartic-like curve with a local maximum near the origin and two symmetric local minima. The central peak occurs around \((0, 1)\text{.}\) The valleys are near \(x \approx -2\) and \(x \approx 2\) with values around \(y \approx -4\text{.}\) As \(|x|\) increases beyond these minima, the function rises steeply on both sides.
    Figure 1.1.4. Graph of \(g(x)\text{.}\)
    🔗
    🔗
    🔗
  3. The function \(k(x)=-2x^2+8x-7\text{.}\)
    🔗
    🔗
  4. The function \(q(x)=x^3-x\text{.}\)
    🔗
    🔗
🔗
🔗
Complete all the information for each function below.
🔗
Function: \(f(x)=x\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.5. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=c\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.6. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=x^2\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.7. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=\sqrt{x}\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.8. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=x^3\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.9. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=\sqrt[3]{x}\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.10. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=|x|\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.11. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(\displaystyle{f(x)=\frac{1}{x}}\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.12. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=2^x\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.13. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
Function: \(f(x)=\log_2{x}\)
🔗
Name:
🔗
A diagram showing two components side by side. On the left is a blank coordinate grid with horizontal and vertical axes marked with arrows, ready for plotting points. On the right is a two-column table labeled \(x\) and \(f(x)\text{,}\) with five empty rows for entering values. This layout is designed for recording function values in the table and graphing them on the grid.
Figure 1.1.14. Blank graph and table for plotting \(f(x)\text{.}\)
🔗
Domain:
🔗
Range:
🔗
Even/Odd:
🔗
🔗
PrevTopNext