Some functions evaluate to indeterminate value at some input values. This is illustrated with an example.

*click on the content to continue..*

What is the value of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=2.2`?

- `0/0`
- indeterminate value
- `3.2`
- `3.2`

The answer is '`3.2`'. On substituting `x=2.2`, we get `f(2.2)= color(deepskyblue)(2.2^2-1)/color(coral)(2.2-1)`.

What is the value of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=1`?

- `0/0`
- indeterminate value
- both the above
- both the above

The answer is 'Both the above'. On substituting `x=1`, we get

`f(1)`

`quad quad =color(deepskyblue)(1^2-1)/color(coral)(1-1)`

`quad quad = color(deepskyblue)(0)/color(coral)(0)`.

Let us closely examine the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`. Can the numerator be factorized?

- Yes, `x^2-1 = (x+1)(x-1)`
- Yes, `x^2-1 = (x+1)(x-1)`
- Cannot be factorized as Factorization is only for numbers, not for algebraic expressions

The answer is 'Yes - `x^2-1 = (x+1)(x-1)`'.

Rewriting the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` as the function `f(x) = color(deepskyblue)((x+1)(x-1))/color(coral)(x-1)`. Can the function be simplified?

- Simplified to `f(x)= x+1`
- Simplified only when `x!=1` to `f(x)= x+1`
- Simplified only when `x!=1` to `f(x)= x+1`

The answer is 'Simplified to `f(x)= x+1` when `x!=1`'. Note that `0` cannot be canceled out in expressions or equations.

So the given function

`f(x)`

`quad quad = x+1 ` when `x!=1`

`quad quad = (x^2-1)/(x-1) ` when `x=1`

By this it is concluded that `f(x)|_(x=1)` is indeterminate value `0/0`.

Many students wrongly understand that the algebraic simplification (like canceling `x-1` in the example above) solves the indeterminate value. It is not so -- the function remains indeterminate at that input value `x=1`.

** Function evaluates to indeterminate value: ** Function `f(x)` evaluates to indeterminate value for `x=a` if `f(a) = 0/0`.

*Solved Exercise Problem: *

Given `f(x)=(x^3-8)/(x^2-4)` What is `f(2)`?

- `0/0`
- `0/0`
- `10`
- `1`

The answer is '`0/0`'

*switch to slide-show version*