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Which of the following represents the difference quotient for f of x is equal to 4 over x question mark

the quantity 4 times x plus h end quantity over the quantity x times h times the quantity x plus h end quantity
negative of the quantity 4 times x minus h end quantity over the quantity x times h times the quantity x plus h end quantity
4 over the quantity x times the quantity x plus h end quantity
negative 4 over the quantity x times the quantity x plus h end quantity

Sagot :

MrRoyal

The difference quotient is simply how the average rate of change compares over an interval.

The difference quotient of [tex]\mathbf{f(x) = \frac 4x}[/tex] is [tex]\mathbf{-\frac{4}{x + h}}[/tex]

The function is given as:

[tex]\mathbf{f(x) = \frac 4x}[/tex]

The difference quotient for f(x) is calculated using:

[tex]\mathbf{Difference\ quotient = \frac{f(x + h) - f(x)}{h}}[/tex]

We have:

[tex]\mathbf{f(x) = \frac 4x}[/tex]

Substitute x + h for x

[tex]\mathbf{f(x + h) = \frac 4{x + h}}[/tex]

The difference quotient becomes

[tex]\mathbf{Difference\ quotient = \frac{f(x + h) - f(x)}{h}}[/tex]

[tex]\mathbf{Difference\ quotient = \frac{\frac{4}{x + h} - \frac{4}{x}}{h}}[/tex]

Take LCM

[tex]\mathbf{Difference\ quotient = \frac{\frac{4x - 4x - 4h}{x + h}}{h}}[/tex]

[tex]\mathbf{Difference\ quotient = \frac{\frac{- 4h}{x + h}}{h}}[/tex]

Rewrite as:

[tex]\mathbf{Difference\ quotient = \frac{- 4h}{x + h} \div h}[/tex]

Express as products

[tex]\mathbf{Difference\ quotient = \frac{- 4h}{x + h} \times \frac 1h}[/tex]

[tex]\mathbf{Difference\ quotient = \frac{- 4}{x + h}}[/tex]

[tex]\mathbf{Difference\ quotient = -\frac{4}{x + h}}[/tex]

Hence, the difference quotient of [tex]\mathbf{f(x) = \frac 4x}[/tex] is [tex]\mathbf{-\frac{4}{x + h}}[/tex]

Read more about difference quotient at:

https://brainly.com/question/2581441

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