7.3 – Power Functions & Function Operations

Operations on Functions: for any two functions f(x) & g(x) 1. Addition h(x) = f(x) + g(x) 2. Subtraction h(x) = f(x) – g(x) 3. Multiplication h(x) = f(x) · g(x) OR f(x)g(x) 4. Division h(x) = f(x) / g(x) OR f(x) ÷ g(x)

Something New… Domain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.) Domain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

Ex: Let f(x) = 3x 1/3 & g(x) = 2x 1/3. Find (A) f(x) + g(x), (B) f(x) – g(x), and (C) the domain for each. A. 3x 1/3 + 2x 1/3 = 5x 1/3 = 5x 1/3 B. 3x 1/3 – 2x 1/3 = x 1/3 = x 1/3 C. Domain of (a) all real numbers Domain of (b) all real numbers

Ex: Let f(x) = x² + 4x – 3 and g(x) = x² - 1. Find (A) f(x) + g(x), (B) f(x) – g(x), (C) g(x) – f(x), and (d) the domain for each. A. 2x² + 4x – 4 Domain: All real #’s B. 4x – 2 Domain: All real #’s C. -4x + 2 Domain: All real #’s

Note: Note: The domain of the resulting function is determined by the functions being used, not just the resulting function itself. Example: Domain of (f - g) is all non-negative real numbers. Domain of (f /g) is all positive real numbers.

Ex: Let f(x) = 4x 1/3 & g(x) = x 1/2. Find (A) the product, (B) the quotient, and (C) the domain for each. A. 4x 1/3 · x 1/2 = 4x 1/3+1/2 = 4x 5/6 B. B. = 4x 1/3-1/2 = 4x 1/3-1/2 = 4x -1/6 = 4x -1/6 = C. C. Domain of (A) all reals ≥ 0, because you can’t take the 6 th root of a negative number. Domain of (B) all reals > 0, because you can’t take the 6 th root of a negative number and you can’t have a denominator of zero.

Ex: Let f(x) = 3x and g(x) = x 1/4. Find (A) f(x) · g(x), (B) f(x) ÷ g(x), and (C) the domain for each. A. 3x 5/4 Domain: All positive real #’s B. 3x 3/4 Domain: All positive real #’s

Homework Power Functions! pgs #32-35, even Quiz on Thursday, February 14 th

7.3 – Power Functions & Function Operations (Day 2)

Objectives: Today we will… Today we will… –Clarify and expand our understanding of finding domains. –Be introduced to the composition of functions. –Explain our reasoning behind why we solve problems a certain way.

Operations on Functions: for any two functions f(x) & g(x) 1. Addition h(x) = f(x) + g(x) 2. Subtraction h(x) = f(x) – g(x) 3. Multiplication h(x) = f(x) · g(x) OR f(x)g(x) 4. Division h(x) = f(x) / g(x) OR f(x) ÷ g(x) 5. Composition h(x) = f(g(x)) or h(x) = g(f(x)) ** Domain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

REVIEW: REVIEW: The domain of the resulting function is determined by the functions being used, not just the resulting function itself. Example: Domain of (f - g) is all non-negative real numbers. Domain of (f /g) is all positive real numbers.

Composition f(g(x)) means you take the function g and plug it in for the x-values in the function f, then simplify. f(g(x)) means you take the function g and plug it in for the x-values in the function f, then simplify. g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify. g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify.

Ex: Let f(x) = 2x -1 & g(x) = x 2 – 1. Find (A) f(g(x)), (B) g(f(x)), (C) f(f(x)), and (d) the domain of each. A. 2(x 2 – 1) -1 = B. (2x -1 ) 2 – 1 = 2 2 x -2 – 1 = C. 2(2x -1 ) -1 = 2(2 -1 x) = D. Domain of (A) all reals except x = ±1. Domain of (B) all reals except x = 0. Domain of (C) all reals except x = 0 because 2x -1 can’t have x = 0.

Ex: Let f(x) = 3x -1 & g(x) = 2x – 1. Find (A) f(g(x)), (B) g(f(x)), (C) f(f(x)), and (d) the domain of each. A. 3(2x – 1) -1 = B. 2(3x -1 ) – 1 = 6x -1 – 1 = C. 3(3x -1 ) -1 = 3(3 -1 x) = D. Domain of (A) all reals except x = -1/2. Domain of (B) all reals except x = 0. Domain of (C) all reals except x = 0 because 3x -1 can’t have x = 0.

Homework pgs #36-39, Quiz on Thursday, February 14 th Power Functions!