## How to Multiply and Divide Functions

Simply carry out the instructions for the operation to add, subtract, multiply, or divide. The limitations of the two functions that went into creating the new function will be present in its domain. Divide has the additional rule that the function by which we are dividing cannot be zero.
We can multiply and divide functions just like we can multiply and divide numbers. If we had functions $$f$$ and $$g$$, we could make two new functions: $$f \times g$$ and $$\frac{f}{g}$$ .

Look at these examples to see how to multiply and divide functions:

### Example1

Consider: $$f \ = \ 5x \ + \ 8$$, $$g \ = \ x \ - \ 4$$, Find: $$(f \times g)(4)$$

Solution:

First, we just do as the operation says: $$(f \times g)(x) \ = \ (5x \ + \ 8) \times (x \ - \ 6) \ = \ 5x^2 \ - \ 12x \ - \ 48$$

Now, substitute $$x$$ with $$4$$: $$(f \times g)(4) \ = \ 5(4)^2 \ - \ 12(4) \ - \ 48 \ = \ -16$$

### Example2

Consider: $$f \ = \ x \ - \ 14$$, $$g \ = \ 3x \ + \ 9$$, Find: $$(\frac{f}{g} \ )(2)$$

Solution:

First, we just do as the operation says: $$(\frac{f}{g} \ )(x) \ = \ \frac{x \ - \ 14}{3x \ + \ 9}$$

Now, substitute $$x$$ with $$2$$: $$(\frac{f}{g} \ )(2) \ = \ \frac{x \ - \ 14}{3x \ + \ 9} \ = \ \frac{2 \ - \ 14}{3(2) \ + \ 9} \ = \ -\frac{12}{15} \ = \ -\frac{4}{5}$$

### Exercises for Multiplying and Dividing Functions

1) $$h(x) = 9x \\ g(x) = 3x \ + \ 3$$
Find $$(h . g)(x)$$

2) $$h(x) = 4x \\ g(x) = -16x^3 \ + \ 8x^2$$
Find $$(\frac{g}{h})(x)$$

3) $$h(x) = -8x \\ g(x) = 2x \ - \ 3$$
Find $$(h . g)(x)$$

4) $$h(x) = 6x \\ g(x) = 18x^3 \ + \ 18x^2$$
Find $$(\frac{g}{h})(3)$$

5) $$h(x) = 11x \\ g(x) = -22x^3 \ + \ 22x^2$$
Find $$(\frac{g}{h})(x)$$

6) $$h(x) = 10x \\ g(x) = 6x \ + \ 4$$
Find $$(h . g)(3)$$

7) $$h(x) = 9x \\ g(x) = -27x^3 \ + \ 27x^2$$
Find $$(\frac{g}{h})(3)$$

8) $$h(x) = 3x \\ g(x) = 6x \ + \ 4$$
Find $$(h . g)(3)$$

9) $$h(x) = 2x \\ g(x) = -4x^3 \ + \ 8x^2$$
Find $$(\frac{g}{h})(3)$$

10) $$h(x) = -9x \\ g(x) = 3x \ - \ 2$$
Find $$(h . g)(x)$$

1) $$h(x) = 9x \\ g(x) = 3x \ + \ 3$$
Find $$(h . g)(x)$$
$$\color{red}{ (h . g)(x) = (9x) \times (3x \ + \ 3) \\ (h . g)(x) = 27x^2 \ + \ 27x}$$
2) $$h(x) = 4x \\ g(x) = -16x^3 \ + \ 8x^2$$
Find $$(\frac{g}{h})(x)$$
$$\color{red}{(\frac{g}{h})(x) = (-16x^3 \ + \ 8x^2) \div (4x) \\ (\frac{g}{h})(x) = -4x^2 \ + \ 2x}$$
3) $$h(x) = -8x \\ g(x) = 2x \ - \ 3$$
Find $$(h . g)(x)$$
$$\color{red}{ (h . g)(x) = (-8x) \times (2x \ - \ 3) \\ (h . g)(x) = -16x^2 \ + \ 24x}$$
4) $$h(x) = 6x \\ g(x) = 18x^3 \ + \ 18x^2$$
Find $$(\frac{g}{h})(3)$$
$$\color{red}{ (\frac{g}{h})(x) = (18x^3 \ + \ 18x^2) \div (6x) \\ (\frac{g}{h})(x) = 3x^2 \ + \ 3x \\ (\frac{g}{h})(3) = 3(3)^2 \ + \ 3(3) \\ (\frac{g}{h})(3) = 36}$$
5) $$h(x) = 11x \\ g(x) = -22x^3 \ + \ 22x^2$$
Find $$(\frac{g}{h})(x)$$
$$\color{red}{(\frac{g}{h})(x) = (-22x^3 \ + \ 22x^2) \div (11x) \\ (\frac{g}{h})(x) = -2x^2 \ + \ 2x}$$
6) $$h(x) = 10x \\ g(x) = 6x \ + \ 4$$
Find $$(h . g)(3)$$
$$\color{red}{ (h . g)(x) = (10x) \times (6x \ + \ 4) \\ (h . g)(x) = 60x^2 \ + \ 40x \\ (h.g)(3) = 60(3) ^ 2 \ + \ 40(3) \\ (h.g)(3) = 660}$$
7) $$h(x) = 9x \\ g(x) = -27x^3 \ + \ 27x^2$$
Find $$(\frac{g}{h})(3)$$
$$\color{red}{ (\frac{g}{h})(x) = (-27x^3 \ + \ 27x^2) \div (9x) \\ (\frac{g}{h})(x) = -3x^2 \ + \ 3x \\ (\frac{g}{h})(3) = -3(3)^2 \ + \ 3(3) \\ (\frac{g}{h})(3) = -18}$$
8) $$h(x) = 3x \\ g(x) = 6x \ + \ 4$$
Find $$(h . g)(3)$$
$$\color{red}{ (h . g)(x) = (3x) \times (6x \ + \ 4) \\ (h . g)(x) = 18x^2 \ + \ 12x \\ (h.g)(3) = 18(3) ^ 2 \ + \ 12(3) \\ (h.g)(3) = 198}$$
9) $$h(x) = 2x \\ g(x) = -4x^3 \ + \ 8x^2$$
Find $$(\frac{g}{h})(3)$$
$$\color{red}{ (\frac{g}{h})(x) = (-4x^3 \ + \ 8x^2) \div (2x) \\ (\frac{g}{h})(x) = -2x^2 \ + \ 4x \\ (\frac{g}{h})(3) = -2(3)^2 \ + \ 4(3) \\ (\frac{g}{h})(3) = -6}$$
10) $$h(x) = -9x \\ g(x) = 3x \ - \ 2$$
Find $$(h . g)(x)$$
$$\color{red}{ (h . g)(x) = (-9x) \times (3x \ - \ 2) \\ (h . g)(x) = -27x^2 \ + \ 18x}$$

## Multiplying and Dividing Functions Practice Quiz

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