Sec 2.8 Algebra of Functions and Function Composition PDF

## Section 2.8 ### Algebra of Functions and Function Composition ### Prior to Class Lecture - Read Section 2.8 (pp 262-265, 270) - Be familiar with the Sum, Difference, Product and Quotient of Functions - Be able to find and simplify the difference quotient for a given function - Note how to estima...

## Section 2.8 ### Algebra of Functions and Function Composition ### Prior to Class Lecture - Read Section 2.8 (pp 262-265, 270) - Be familiar with the Sum, Difference, Product and Quotient of Functions - Be able to find and simplify the difference quotient for a given function - Note how to estimate function values from a graph ### Class Lecture #### Questions from Section 2.7??? #### Sum, Difference, Product, and Quotient of Functions If *f(x)* and *g(x)* are functions, then for all *x* common to domains of both functions, the sum, difference, product, and quotient of *f(x)* and *g(x)* are defined as follows: 1. **Sum**: *(f + g)(x) = f(x) + g(x)* 2. **Difference**: *(f – g)(x) = f(x) – g(x)* 3. **Product**: *(fg)(x) = (f(x))(g(x))* 4. **Quotient**: *(f / g)(x) = f(x) / g(x)*, provided *g(x) ≠ 0* If *f(x) = x² + 1* and *g(x) = 3x – 5*, find the following and give the domain in interval notation. 1. *(f + g)(x) = f(x) + g(x)* - * = x² + 1 + 3x – 5* - * = x² + 3x – 4* - domain *(– ∞, ∞)* 2. *(f – g)(x) = f(x) – g(x)* - * = x² + 1 – (3x – 5)* - * = x² + 1 – 3x + 5* - * = x² – 3x + 6* - *(– ∞, ∞)* domain 3. *(fg)(x) = (f(x))(g(x))* - * = (x² + 1)(3x – 5)* - * = 3x³ – 5x² + 3x – 5* - domain *(– ∞, ∞)* 4. *(f / g)(x) = f(x) / g(x)*, *g(x) ≠ 0* - * = x² + 1 / 3x – 5* - domain *(– ∞, 5/3) U (5/3, ∞)* 5. *(f + g)(3) = x² + 3x – 4* - * = 3² + 3(3) – 4* - * = 9 + 9 – 4* - * = 14* 6. *(fg)(4) = (f(x))(g(x))* - * = (4² + 1)(3(4) – 5)* - * = (17)(7)* - * = 119* If *f(x) = x² – 5* and *g(x) = √(6 – x)*, find *(fg)(x)* and give the domain in interval notation. - *(fg)(x) = (f(x))(g(x))* - * = (x² – 5)(√(6 – x))* - * = x²√(6 – x) – 5√(6 – x)* - domain *(– ∞, 6]* Given *m(x) = (x – 4) / (x² – 25)* and *n(x) = (x – 4) / (5 – x)*, find the following and give the domain in interval notation. - *m(x) = (x – 4) / (x² – 25)* - *n(x) = (x – 4) / (5 – x)*, *x ≠ 5* 1. *(m / n)(x) = m(x) / n(x)* - * = ((x – 4)/(x² – 25)) / ((x – 4)/(5 – x))* - * = (x – 4) / (x² – 25) * (5 – x) / (x – 4)* - * = (5 – x) / (x² – 25)* - * = (5 – x) / ((x + 5)(x – 5))* - * = -1 / (x + 5)*, *x ≠ -5, 5, 4* - domain *(– ∞, –5) U (–5, 4) U (4, 5) U (5, ∞)* 2. *(m + n)(x) = (x – 4) / (x² – 25) + (x – 4) / (5 – x)* - * = ((x – 4)/(x² – 25)) + ((x – 4)/(5 – x))* - * = ((x – 4)(5 – x) + (x – 4)(x² – 25)) / ((x² – 25)(5 – x))* - * = (5x – x² – 20 + 4x + x³ – 25x) / ((x² – 25)(5 – x))* - * = (x³ – x² – 16x – 20) / ((x² – 25)(5 – x))* - domain *x≠-5, 5* - *(– ∞, -5) U (-5, 5) U (5, ∞)* Two graphs are shown. Please describe them. * The first graph shows the functions *f(x)* and *g(x)*. * The second graph shows the function *(f + g)(x)*. * The third graph shows the function *(f - g)(x)* * The fourth graph shows the calculation for *(f * g)(-2) = (f(-2))(g(-2))* and *( (f / g)(3) = f(3) / g(3))*. ### Difference Quotient * *f(x) = 4x + 7* * *f(x + h) – f(x) / h* * *f(3) = 4(3) + 7* * * = 12 + 7* * * = 19* * *f(x + h) = 4(x + h) + 7* * * = 4x + 4h + 7* * **f(x) = 4x + 7** * *f(x + h) – f(x) / h = (4x + 4h + 7 – (4x + 7)) / h* * * = (4x + 4h + 7 – 4x – 7) / h* * * = 4h / h* * * = 4, h ≠ 0* * *f(x) = 2x² – 3x + 4* * * f(x + h) – f(x) / h* * *f(x + h) – f(x) = 2(x + h)² – 3(x + h) + 4 – (2x² – 3x + 4) / h* * * = 2(x² + 2xh + h²) – 3x – 3h + 4 – 2x² + 3x – 4 / h* * * = 2x² + 4xh + 2h² – 3x – 3h + 4 – 2x² + 3x – 4 / h* * * = 4xh + 2h² – 3h / h* * * = h(4x + 2h – 3) / h* * * = 4x + 2h – 3, h≠ 0* Chapter 2, Section 8: Lecture: Introduction to Difference Quotient