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For each of the following, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain.

For each property, write inCreasing, Decreasing, Even, Odd, inVertible in that order (alphabetical). For example, if the function is increasing, odd, and invertible, submit "COV". If the function is none of the above, submit "NONE".

(a) \(x^2-2x+3\)

(b) \(\sqrt{x-5} \)

(c) \(\frac{x}{x^2}+1\)

(d) \(x+1+\frac1x\)

(e) \(|x|-\sqrt x\)

 Jan 12, 2019

what do you mean by increasing and decraesing?

 Jan 12, 2019

increasing is when the gradient of the tangent is positive

decreasing is when the gradient of the tangent is negative

Melody  Jan 12, 2019

What does 'invertible' mean?

 Jan 12, 2019

it means one-to-one, like for every x only one y

 Jan 12, 2019

Thanks asdf :)


I expect it means more than that.

It probably means every x has just one y  AND every y has just one x.

So that it is a function for all values of x and its inverse is also a function for all values of x.

I quickly looked it up. That seems to be what it means.

Melody  Jan 12, 2019
edited by Melody  Jan 12, 2019



It is increasing when the gradient of the tangent is positive    i.e.       \(\frac{dy}{dx}>0\)


It is decreasing when the gradient of the tangent is negative    i.e.       \(\frac{dy}{dx}<0\)


It is even if it is symmetrical around the y axis, which is the line x=0    SO      \(f(-x)=f(x)\)


It is odd if it is has 180 degree rotational symmetrical around the (0,0), SO      \(f(-x)=-f(x)\)


It is invertible is both the original and the inverse are functions. This happens when:

It is invertible is the gradient is always increaing or always decreasing, 0 is ok too. 

     \(\frac{dy}{dx}\ge0\quad or \quad\frac{dy}{dx}\le0 \qquad \text{ for all real values of x}\)  



Now you can do them yourself :)

 Jan 13, 2019

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