Math Problem Statement

pls answer this with solution

pls answer this with solution

Solution

To solve this problem, let's go through each part step by step, based on the functions provided in the image.

Given Functions:

  1. f(x)=x+3f(x) = x + 3
  2. g(x)=2x5g(x) = 2x - 5
  3. h(x)=2x2+3x+2h(x) = 2x^2 + 3x + 2
  4. p(x)=3x23x+7p(x) = 3x^2 - 3x + 7
  5. t(x)=x29t(x) = x^2 - 9

Problems to Solve:

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

(f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x) =(x+3)+(2x5)= (x + 3) + (2x - 5) =x+2x+35= x + 2x + 3 - 5 =3x2= 3x - 2

b. (tg)(x)(t - g)(x)

(tg)(x)=t(x)g(x)(t - g)(x) = t(x) - g(x) =(x29)(2x5)= (x^2 - 9) - (2x - 5) =x292x+5= x^2 - 9 - 2x + 5 =x22x4= x^2 - 2x - 4

c. (hg)(x)(h \cdot g)(x)

(hg)(x)=h(x)g(x)(h \cdot g)(x) = h(x) \cdot g(x) =(2x2+3x+2)(2x5)= (2x^2 + 3x + 2) \cdot (2x - 5)

Expanding the multiplication: =2x22x+2x2(5)+3x2x+3x(5)+22x+2(5)= 2x^2 \cdot 2x + 2x^2 \cdot (-5) + 3x \cdot 2x + 3x \cdot (-5) + 2 \cdot 2x + 2 \cdot (-5) =4x310x2+6x215x+4x10= 4x^3 - 10x^2 + 6x^2 - 15x + 4x - 10 =4x34x211x10= 4x^3 - 4x^2 - 11x - 10

d. (f+h)(x)(f + h)(x)

(f+h)(x)=f(x)+h(x)(f + h)(x) = f(x) + h(x) =(x+3)+(2x2+3x+2)= (x + 3) + (2x^2 + 3x + 2) =2x2+3x+x+3+2= 2x^2 + 3x + x + 3 + 2 =2x2+4x+5= 2x^2 + 4x + 5

e. fp(x)\frac{f}{p}(x)

fp(x)=f(x)p(x)\frac{f}{p}(x) = \frac{f(x)}{p(x)} =x+33x23x+7= \frac{x + 3}{3x^2 - 3x + 7}

This is the simplified form since the numerator and the denominator have no common factors.

f. gf(x)\frac{g}{f}(x)

gf(x)=g(x)f(x)\frac{g}{f}(x) = \frac{g(x)}{f(x)} =2x5x+3= \frac{2x - 5}{x + 3}

This is the simplified form since the numerator and the denominator have no common factors.

Summary of Answers:

  1. (f+g)(x)=3x2(f + g)(x) = 3x - 2
  2. (tg)(x)=x22x4(t - g)(x) = x^2 - 2x - 4
  3. (hg)(x)=4x34x211x10(h \cdot g)(x) = 4x^3 - 4x^2 - 11x - 10
  4. (f+h)(x)=2x2+4x+5(f + h)(x) = 2x^2 + 4x + 5
  5. fp(x)=x+33x23x+7\frac{f}{p}(x) = \frac{x + 3}{3x^2 - 3x + 7}
  6. gf(x)=2x5x+3\frac{g}{f}(x) = \frac{2x - 5}{x + 3}

Would you like more details or have any other questions?

Further Questions:

  1. What are the domain restrictions for fp(x)\frac{f}{p}(x)?
  2. How do you find the inverse of f(x)f(x)?
  3. What is the composition of f(g(x))f(g(x))?
  4. How do the graphs of f(x)f(x) and g(x)g(x) compare?
  5. What are the critical points of h(x)h(x)?

Tip: When dealing with operations on functions, always ensure that the domain of the resulting function considers the domains of the individual functions involved.

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Operations
Polynomial Multiplication
Division of Functions

Formulas

Addition of functions: (f + g)(x) = f(x) + g(x)
Subtraction of functions: (t - g)(x) = t(x) - g(x)
Multiplication of functions: (h * g)(x) = h(x) * g(x)
Division of functions: (f/p)(x) = f(x) / p(x)

Theorems

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Suitable Grade Level

Grades 9-12