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)

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

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

This is simply adding the two functions:
\[
(f + g)(x) = f(x) + g(x)
\]
\[
f(x) = 3x + 2x^2 - 10
\]
\[
g(x) = 5x - x^2 + 15
\]
Adding them together:
\[
(f + g)(x) = (3x + 2x^2 - 10) + (5x - x^2 + 15)
\]
\[
= 3x + 5x + 2x^2 - x^2 - 10 + 15
\]
\[
= 8x + x^2 + 5
\]

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

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

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

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

### C. \((f - g)(x)\)

Subtract the functions:
\[
(f - g)(x) = f(x) - g(x)
\]
\[
= (3x + 2x^2 - 10) - (5x - x^2 + 15)
\]
\[
= 3x - 5x + 2x^2 - (-x^2) - 10 - 15
\]
\[
= -2x + 3x^2 - 25
\]

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

### D. \((f - g)(-6)\)

Now, substitute \( x = -6 \) into \((f - g)(x)\):
\[
(f - g)(-6) = 3(-6)^2 - 2(-6) - 25
\]
\[
= 3(36) + 12 - 25
\]
\[
= 108 + 12 - 25
\]
\[
= 95
\]

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

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

Multiply the functions:
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
\[
= (3x + 2x^2 - 10) \cdot (5x - x^2 + 15)
\]
We can expand this product using distribution:
\[
= (3x)(5x) + (3x)(-x^2) + (3x)(15) + (2x^2)(5x) + (2x^2)(-x^2) + (2x^2)(15) + (-10)(5x) + (-10)(-x^2) + (-10)(15)
\]
\[
= 15x^2 - 3x^3 + 45x + 10x^3 - 2x^4 + 30x^2 - 50x + 10x^2 - 150
\]
\[
= -2x^4 + 7x^3 + 55x^2 - 5x - 150
\]

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

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

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

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

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

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

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

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

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

---

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 \( (f + g)(x) \) and \( (f - g)(x) \) to compare their shapes?
4. What happens when \( g(x) = 0 \) in \( (f / g)(x) \)?
5. How do you differentiate and integrate the product of two functions?

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

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).

Learn how to perform operations on functions, including addition, subtraction, multiplication, and division of the polynomial functions f(x) = 3x + 2x^2 - 10 and g(x) = 5x - x^2 + 15. Includes detailed solutions for (f + g)(x), (f - g)(x), (f * g)(x), and (f / g)(x), and the evaluation at specific values of x. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation