Math 108 Lecture 14 - Combining Functions

Math 108 Lecture 14 Notes

1.9 Combining Functions

1.9.1 Arithmetic with Functions

• Definition: Let 𝑓 and 𝑔 be functions that share the same domain. The following operations are defined:
  • Sum of Functions: The sum of 𝑓 and 𝑔 is the function 𝑓 + 𝑔 defined by
    $(f + g)(x) = f(x) + g(x)$
  • Difference of Functions: The difference of 𝑓 and 𝑔 is the function 𝑓 − 𝑔 defined by
    $(f - g)(x) = f(x) - g(x)$
  • Product of Functions: The product of 𝑓 and 𝑔 is the function 𝑓 ⋅ 𝑔 defined by
    $(f imes g)(x) = f(x) imes g(x)$
  • Quotient of Functions: The quotient of 𝑓 and 𝑔 is the function
    $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ such that 𝑔(𝑥) ≠ 0.

1.9.3 Piecewise Functions

• Definition: Functions that are comprised of different functions, depending on the domain, are called piecewise-defined functions.
  • Example of a Piecewise Function:
    $A(x) = |x|$
• Example Problem: Sketch the graph of the following piecewise-defined function:
  f(x) = \begin{cases}
  2x^2, & x \leq -2 \
  x + 1, & -2 < x < 4 \
  -3, & x \geq 4
  \end{cases}

1.9.4 Activity on Piecewise Functions

• In this segment, we will explore two different piecewise functions entirely by hand based on familiar properties of linear and quadratic functions.

a. Function Definition of p

• Function p: p(x) = \begin{cases}
  • (x + 2)^2 + 2, & x < 0 \
    \frac{1}{2}(x - 2)^2 + 1, & x \geq 0
    \end{cases}

Evaluating p at Specific Points

• Values to Find: Determine the values of
  • p(-4)
  • p(-2)
  • p(0)
  • p(2)
  • p(4)

b. Vertex of Quadratic Parts

• Quadratic part for x < 0: Find the vertex of the quadratic portion,
  $-(x + 2)^2 + 2$
• Quadratic part for x ≥ 0: Find the vertex of the quadratic portion,
  $\frac{1}{2}(x - 2)^2 + 1$

c. Zero Values and Y-Intercept

• Values where p(x) = 0: Identify the values of 𝑥 for which 𝑝(𝑥) = 0.
• Y-Intercept: Determine the y-intercept of function p.

d. Sketching the Graph of y = p(x)

• Task: Sketch an accurate and labeled graph of $y = p(x)$ on the axes provided in Figure 1.9.9.

e. Function f from Graph

• Goal: For the function f defined by Figure 1.9.10, determine a piecewise-defined formula for f that is expressed in bracket notation similar to the definition of $y = p(x)$ above.