Question

When comparing the f(x) = x^2 – x and g(x) = log(2x + 1), on which interval are both functions positive?(–∞, 0) (0, 1) (1, ∞) (∞, ∞)

Answer 1

`(1\text{ , }\infty)`

Step-by-step explanation:

Since for g(-∞), is not a possible answer. we remove it.

We also remove (∞,∞), because it goes back to being simply ∞.

Now for

(0 , 1 )

`\begin{gathered} f(x)\text{ = x}^2\text{ - x } \\ f(1)\text{ = 1}^2\text{ - 1 = 0} \end{gathered}`

`f(0)\text{ = 0}^2\text{ - 0 = 0}`

This is not positive, nor negative so we will also put this one on the side.

For the last interval (1, ∞)

`\begin{gathered} f(1)\text{ = 0 } \\ f(\infty)\text{ = }\infty^2-\infty\text{ = }\infty \end{gathered}`

=> f(x) is positive

`g(1)\text{ = log\lparen2*1 + 1\rparen = log\lparen3\rparen }\approx\text{ 0.477712}`

g(1) is positive

`g(\infty)\text{ = log\lparen2*}\infty+1)\text{ = log\lparen}\infty)\text{ = }\infty`

=> g(∞) is positive

Hence f(x) and g(x) are both positve on the interval (1, 8)

NB:

Technically you should say g(x) tend to infinity (or a approaches infinity) not is equal to infinity, because infinity is not a number