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# MATH121, Lesson 4.5 Notes Math 121

OleMiss

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This 3 page Class Notes was uploaded by Mallory McClurg on Sunday September 25, 2016. The Class Notes belongs to Math 121 at University of Mississippi taught by Dirle in Fall 2016. Since its upload, it has received 16 views. For similar materials see College Algebra in Math at University of Mississippi.

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Date Created: 09/25/16

MATH121 CHHA APPTTEER R 4 NOOT TEESS Lesson 4.5 – Combining Functions EXAMPLE 1. Find (f ° g)(2), when f(x) = x + 1 and g(x) = x + x . 3 2 (First, we need to find the function g(2).) g(2) = 2 + 2 = 12 (Now, substitute the value 12 in for x in the function f(x).) f(12) = 12 + 1 = 145 (f ° g)(2) = 145 (This is our answer!) EXAMPLE 2. 3 3/2 Find the formula for (f + g)(x), when f(x) = √x and g(x) = x , then find the domain. (First, we need to add the functions f(x) and g(x).) 3 3/2 (f + g) (x) = √x + x (Pretty simple. Now, we need to find the domain of this function. Are there any numbers we know of that we can take the cube root of that will be undefined? No… But what about x 3/2 ? This part of the problem is only defined/real when x is greater than or equal to zero. Try it out. (-1) 3/2 is –i, Totally not a real number.) Domain: [0, ∞) (This is our answer! X can equal to any number greater than zero, to keep this function defined.) EXAMPLE 3. Find the formula for (f ° g)(x), when f(x) = x – 3 and g(x) = x , then find the 3 domain. (To find f(g(x)), we need to substitute g( x) into the f(x) function.) (f °g)(x) = x – 3 (Now, to find the domain, ask yourself, what values of x would make this undefined? Now, tell yourself…Nothing.) Domain: (-∞,∞) or ALL REAL NUMBERS Now, find the formula for (g ° f)(x) for the same two functions. (Substitute f(x) into the g(x) function.) 3 (g ° f)(x) = (x – 3) (Now, to find the domain, ask yourself…what value for x would make this undefined? Nothing again, because try subtracting 3 from any number and raising it to the third power and see if it’s a real number. It will be.) Domain: (-∞,∞) or ALL REAL NUMBERS EXAMPLE 4. Find the formula for (f ° g)(x), when f(x) = √(x – 3) and g(x) = x – 2. (f ° g)(x) = √((x – 2) – 3) = √(x – 5) Now, find the formula for (g ° f)(x) for the same functions. (g ° f)(x) = √(x – 3) – 2 MATH121 CHHA APPTTEERR 4 NO OTTEESS EXAMPLE 5. Find (f ° g)(9), when f(x) = 3x + 1 and g(x) = x 3/2 + 5. (First, we need to find g(9) before using that value to substitute into the f(x) function.) g(9) = (9) 3/2 + 5 = 32 f(32) = 3(32) + 1 = 97 (f ° g)(9) = 97 (This is our answer!) EXAMPLE 6. Find the formula for (f ° g)(x), when f(x) = (1/x) and g(x) = x – 3, then find the domain. (First, substitute the function g(x) where you see every x in the function f(x).) (f °g)(x) = 1/(x – 3) (Now, we need to find the domain. What numbers can x NOT equal, if we want the fraction to be real/defined? Every number except 3 itself!) Domain: (-∞, 3) U (3, ∞) Now, find the formula for (g ° f)(x) for those same functions. Then find the domain. (g ° f)(x) = (1/x) – 3 (Now, to find the domain, we need to ask ourselves, what value of x would keep this function real/defined? All numbers except zero.) Domain: (-∞, 0) U (0, ∞) EXAMPLE 7. When f(-2) = 4 and g(-2) = -13, … Find (f + g)(-2). (For all of these, we’re adding, subtracting, multiplying and dividing 4 and -13. Super easy.) (f + g)(-2) = 4 + (-13) = -9 Find (f – g)(-2). (f – g)(-2) = 17 Find (f g)(-2). (f g)(-2) = -52 Find (f/g)(-2). (f/g)(-2) = (-4/13) MATH121 CH HAAP PTTEER R 4 NO OTTE ESS EXAMPLE 8. 3 Find the formula for (f + g)(x), when f(x) = √(x + 2) and g(x) = √(x), then find the domain. (First, we need to just add the functions together and simplify as much as we can. ) 3 (f + g)(x) = √(x + 2) + √(x) (Now, we need to find the domain. This is tricky because we have things in both the numerator and the denominator that could make this function undefined. The variable in the denominator must be greater than zero, first of all. And in the numerator, the value of the variable can only be between -2 and 0, including -2, but not including zero itself.) Domain: [-2, 0) U (0, ∞) EXAMPLE 9. Find the formula (f ° g)(x), when f(x) = √(x + 2) and g(x) = (x + 2)/3, then find the domain. (f ° g)(x) = √((x + 2)/3) + 2) (Now, to find the domain, we need to find the non - negative values for what’s under the square root symbol.) ((x + 2)/3) + 2 ≥ 0 (Multiply both sides by 3 to get rid of the fraction.) x + 2 + 6 ≥ 0 x ≥ -8 (So, we know that the function is defined everywhere that x is greater than or equal to -8.) Domain: [-8, ∞) Now, find the formula for (g ° f)(x) for those same functions. Then find the domain. (g ° f)(x) = (√(x + 2) + 2)/3 (Now, to find the domain, we need to determine what non-negative numbers will keep the function defined. So look at the square root symbol; everything under it must be greater than or equal to zero.) x + 2 ≥ 0 x ≥ -2 (Use this to write the domain.) Domain: [-2, ∞)

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