Math Problem Statement

Given the functions f(x) = x + 5, g(x) = 2x - 1, and h(x) = 2x^2 + 9x - 5, determine the following functions: (f + g)(x), (f - g)(x), (f * g)(x), (f / g)(x), (f + g)(3), (f - g)(3), (f * g)(3), (f / g)(3).

Solution

The problem asks us to work with three given functions and perform operations on them. The functions provided are:

  • f(x)=x+5f(x) = x + 5
  • g(x)=2x1g(x) = 2x - 1
  • h(x)=2x2+9x5h(x) = 2x^2 + 9x - 5

We need to determine the following:

a. (f+g)(x)(f + g)(x)

This represents the sum of the functions f(x)f(x) and g(x)g(x).

(f+g)(x)=f(x)+g(x)=(x+5)+(2x1)=3x+4(f + g)(x) = f(x) + g(x) = (x + 5) + (2x - 1) = 3x + 4

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

This represents the difference of the functions f(x)f(x) and g(x)g(x).

(fg)(x)=f(x)g(x)=(x+5)(2x1)=x+6(f - g)(x) = f(x) - g(x) = (x + 5) - (2x - 1) = -x + 6

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

This represents the product of the functions f(x)f(x) and g(x)g(x).

(fg)(x)=f(x)×g(x)=(x+5)(2x1)(f \cdot g)(x) = f(x) \times g(x) = (x + 5)(2x - 1) Expanding this: (fg)(x)=2x2+10xx5=2x2+9x5(f \cdot g)(x) = 2x^2 + 10x - x - 5 = 2x^2 + 9x - 5

d. (fg)(x)\left(\frac{f}{g}\right)(x)

This represents the division of the functions f(x)f(x) and g(x)g(x).

(fg)(x)=f(x)g(x)=x+52x1\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x + 5}{2x - 1}

e. (f+g)(3)(f + g)(3)

Substitute x=3x = 3 into (f+g)(x)=3x+4(f + g)(x) = 3x + 4.

(f+g)(3)=3(3)+4=9+4=13(f + g)(3) = 3(3) + 4 = 9 + 4 = 13

f. (fg)(3)(f - g)(3)

Substitute x=3x = 3 into (fg)(x)=x+6(f - g)(x) = -x + 6.

(fg)(3)=(3)+6=3+6=3(f - g)(3) = -(3) + 6 = -3 + 6 = 3

g. (fg)(3)(f \cdot g)(3)

Substitute x=3x = 3 into (fg)(x)=2x2+9x5(f \cdot g)(x) = 2x^2 + 9x - 5.

(fg)(3)=2(3)2+9(3)5=2(9)+275=18+275=40(f \cdot g)(3) = 2(3)^2 + 9(3) - 5 = 2(9) + 27 - 5 = 18 + 27 - 5 = 40

h. (fg)(3)\left(\frac{f}{g}\right)(3)

Substitute x=3x = 3 into (fg)(x)=x+52x1\left(\frac{f}{g}\right)(x) = \frac{x + 5}{2x - 1}.

(fg)(3)=3+52(3)1=861=85\left(\frac{f}{g}\right)(3) = \frac{3 + 5}{2(3) - 1} = \frac{8}{6 - 1} = \frac{8}{5}


Would you like further details on any of these calculations or additional practice questions? Here are 5 questions to expand your understanding:

  1. What is the result of (f+h)(x)(f + h)(x)?
  2. Find the derivative of (fg)(x)(f \cdot g)(x).
  3. Evaluate h(2)h(2).
  4. Find the inverse of g(x)=2x1g(x) = 2x - 1.
  5. Solve for xx when (fg)(x)=0(f - g)(x) = 0.

Tip: When performing operations on functions, remember to simplify each step as much as possible before substituting specific values.

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Math Problem Analysis

Mathematical Concepts

Function Operations
Substitution
Polynomial Functions
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

Basic Function Operations
Substitution

Suitable Grade Level

Grades 9-12