Math Problem Statement

Operations on Function

• If f(x) = 3x + 2x ^ 2 - 10 and g(x) = 5x - x ^ 2 + 15 find:

A. (f+g)(x)

B. (f + g)(5)

C. (f-g)(x)

D. (f - g)(- 6)

E. (f*g)(x)

F. (f*g)(2)

G. (f/g)(x)

H. (f / g)(3)

Solution

Let's solve the given operations step by step for the functions f(x)=3x+2x210f(x) = 3x + 2x^2 - 10 and g(x)=5xx2+15g(x) = 5x - x^2 + 15.

A. (f+g)(x)(f + g)(x)

This is simply adding the two functions: (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x) f(x)=3x+2x210f(x) = 3x + 2x^2 - 10 g(x)=5xx2+15g(x) = 5x - x^2 + 15 Adding them together: (f+g)(x)=(3x+2x210)+(5xx2+15)(f + g)(x) = (3x + 2x^2 - 10) + (5x - x^2 + 15) =3x+5x+2x2x210+15= 3x + 5x + 2x^2 - x^2 - 10 + 15 =8x+x2+5= 8x + x^2 + 5

So, (f+g)(x)=x2+8x+5(f + g)(x) = x^2 + 8x + 5.

B. (f+g)(5)(f + g)(5)

Now, substitute x=5x = 5 into (f+g)(x)(f + g)(x): (f+g)(5)=52+8(5)+5(f + g)(5) = 5^2 + 8(5) + 5 =25+40+5= 25 + 40 + 5 =70= 70

So, (f+g)(5)=70(f + g)(5) = 70.

C. (fg)(x)(f - g)(x)

Subtract the functions: (fg)(x)=f(x)g(x)(f - g)(x) = f(x) - g(x) =(3x+2x210)(5xx2+15)= (3x + 2x^2 - 10) - (5x - x^2 + 15) =3x5x+2x2(x2)1015= 3x - 5x + 2x^2 - (-x^2) - 10 - 15 =2x+3x225= -2x + 3x^2 - 25

So, (fg)(x)=3x22x25(f - g)(x) = 3x^2 - 2x - 25.

D. (fg)(6)(f - g)(-6)

Now, substitute x=6x = -6 into (fg)(x)(f - g)(x): (fg)(6)=3(6)22(6)25(f - g)(-6) = 3(-6)^2 - 2(-6) - 25 =3(36)+1225= 3(36) + 12 - 25 =108+1225= 108 + 12 - 25 =95= 95

So, (fg)(6)=95(f - g)(-6) = 95.

E. (fg)(x)(f \cdot g)(x)

Multiply the functions: (fg)(x)=f(x)g(x)(f \cdot g)(x) = f(x) \cdot g(x) =(3x+2x210)(5xx2+15)= (3x + 2x^2 - 10) \cdot (5x - x^2 + 15) We can expand this product using distribution: =(3x)(5x)+(3x)(x2)+(3x)(15)+(2x2)(5x)+(2x2)(x2)+(2x2)(15)+(10)(5x)+(10)(x2)+(10)(15)= (3x)(5x) + (3x)(-x^2) + (3x)(15) + (2x^2)(5x) + (2x^2)(-x^2) + (2x^2)(15) + (-10)(5x) + (-10)(-x^2) + (-10)(15) =15x23x3+45x+10x32x4+30x250x+10x2150= 15x^2 - 3x^3 + 45x + 10x^3 - 2x^4 + 30x^2 - 50x + 10x^2 - 150 =2x4+7x3+55x25x150= -2x^4 + 7x^3 + 55x^2 - 5x - 150

So, (fg)(x)=2x4+7x3+55x25x150(f \cdot g)(x) = -2x^4 + 7x^3 + 55x^2 - 5x - 150.

F. (fg)(2)(f \cdot g)(2)

Substitute x=2x = 2 into (fg)(x)(f \cdot g)(x): (fg)(2)=2(2)4+7(2)3+55(2)25(2)150(f \cdot g)(2) = -2(2)^4 + 7(2)^3 + 55(2)^2 - 5(2) - 150 =2(16)+7(8)+55(4)10150= -2(16) + 7(8) + 55(4) - 10 - 150 =32+56+22010150= -32 + 56 + 220 - 10 - 150 =84= 84

So, (fg)(2)=84(f \cdot g)(2) = 84.

G. (f/g)(x)(f / g)(x)

This is dividing the functions: (f/g)(x)=f(x)g(x)=3x+2x2105xx2+15(f / g)(x) = \frac{f(x)}{g(x)} = \frac{3x + 2x^2 - 10}{5x - x^2 + 15}

H. (f/g)(3)(f / g)(3)

Substitute x=3x = 3 into the quotient: f(3)=3(3)+2(3)210=9+1810=17f(3) = 3(3) + 2(3)^2 - 10 = 9 + 18 - 10 = 17 g(3)=5(3)(3)2+15=159+15=21g(3) = 5(3) - (3)^2 + 15 = 15 - 9 + 15 = 21 (f/g)(3)=1721(f / g)(3) = \frac{17}{21}

So, (f/g)(3)=1721(f / g)(3) = \frac{17}{21}.


Would you like further details on any of these steps? Here are 5 related questions to expand your understanding:

  1. How do you find the composition of two functions?
  2. What is the inverse of a function, and how is it computed?
  3. Can you graph (f+g)(x)(f + g)(x) and (fg)(x)(f - g)(x) to compare their shapes?
  4. What happens when g(x)=0g(x) = 0 in (f/g)(x)(f / g)(x)?
  5. How do you differentiate and integrate the product of two functions?

Tip: Always remember to check for values where g(x)=0g(x) = 0 when performing division of functions, as division by zero is undefined.

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Math Problem Analysis

Mathematical Concepts

Algebra
Operations on Functions
Polynomial Functions
Function Composition
Rational Functions

Formulas

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f * g)(x) = f(x) * g(x)
(f / g)(x) = f(x) / g(x)

Theorems

Distributive Property
Quotient of Functions
Polynomial Multiplication

Suitable Grade Level

Grades 9-12