Math Problem Statement

Given the function f(x)=8x-2 and g(x)=-6x+13 find the following: a. (f-g)(x)= b. (f-g)(3)= c. (fg)(x)= d. (fg)(0)=

Solution

Let's go step by step to solve the given problems based on the functions f(x)=8x2f(x) = 8x - 2 and g(x)=6x+13g(x) = -6x + 13.

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

(fg)(x)=f(x)g(x)(f - g)(x) = f(x) - g(x) Substitute the given expressions for f(x)f(x) and g(x)g(x): (fg)(x)=(8x2)(6x+13)(f - g)(x) = (8x - 2) - (-6x + 13) Simplify the expression: (fg)(x)=8x2+6x13=(8x+6x)+(213)=14x15(f - g)(x) = 8x - 2 + 6x - 13 = (8x + 6x) + (-2 - 13) = 14x - 15 So, (fg)(x)=14x15(f - g)(x) = 14x - 15

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

Substitute x=3x = 3 into (fg)(x)=14x15(f - g)(x) = 14x - 15: (fg)(3)=14(3)15=4215=27(f - g)(3) = 14(3) - 15 = 42 - 15 = 27 So, (fg)(3)=27(f - g)(3) = 27

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

(fg)(x)=f(x)g(x)(f \cdot g)(x) = f(x) \cdot g(x) Substitute the given expressions for f(x)f(x) and g(x)g(x): (fg)(x)=(8x2)(6x+13)(f \cdot g)(x) = (8x - 2) \cdot (-6x + 13) Use the distributive property to multiply: (fg)(x)=8x(6x)+8x132(6x)213(f \cdot g)(x) = 8x \cdot (-6x) + 8x \cdot 13 - 2 \cdot (-6x) - 2 \cdot 13 Simplify: (fg)(x)=48x2+104x+12x26=48x2+116x26(f \cdot g)(x) = -48x^2 + 104x + 12x - 26 = -48x^2 + 116x - 26 So, (fg)(x)=48x2+116x26(f \cdot g)(x) = -48x^2 + 116x - 26

d. (fg)(0)(f \cdot g)(0)

Substitute x=0x = 0 into (fg)(x)=48x2+116x26(f \cdot g)(x) = -48x^2 + 116x - 26: (fg)(0)=48(0)2+116(0)26=26(f \cdot g)(0) = -48(0)^2 + 116(0) - 26 = -26 So, (fg)(0)=26(f \cdot g)(0) = -26

Summary of the answers:

a. (fg)(x)=14x15(f - g)(x) = 14x - 15
b. (fg)(3)=27(f - g)(3) = 27
c. (fg)(x)=48x2+116x26(f \cdot g)(x) = -48x^2 + 116x - 26
d. (fg)(0)=26(f \cdot g)(0) = -26

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions for further practice:

  1. Find (f+g)(x)(f + g)(x) for the same functions f(x)f(x) and g(x)g(x).
  2. Compute (f+g)(2)(f + g)(2).
  3. Find the derivative of (fg)(x)(f \cdot g)(x).
  4. Determine the x-value where (fg)(x)=0(f - g)(x) = 0.
  5. Find the x-intercepts of (fg)(x)(f \cdot g)(x).

Tip: When combining functions, always distribute and combine like terms carefully to avoid errors in signs and coefficients.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Function Operations
Distributive Property

Formulas

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

Theorems

Distributive Property of Multiplication over Addition

Suitable Grade Level

Grades 9-10