# Questions related to Relations and Functions

1. Mar 25, 2015

### Raghav Gupta

1. The problem statement, all variables and given/known data

1. Range of the function $\sqrt {x^2+x+1}$ is equal to?

2.ƒ:R---->R is defined as ƒ(x) = x2 -3x +4, then f -1 (2) is equal to?

2. Relevant equations
NA

3. The attempt at a solution
For the first one tried squaring on both the sides but that does not give linear x in terms of y for finding the range.

For second one, I can directly substitute f(x) as 2 for getting the answer.
But I have a confusion that a function must be injective and surjective for the inverse otherwise it inverse must not exist.

2. Mar 25, 2015

### fourier jr

1. you must have $x^2 + x + 1 \geq 0$ so for what x is that true?

2. I think you could start with $2 = x^2 - 3x + 4$ & go from there

3. Mar 25, 2015

### Raghav Gupta

1. I'm not asking for domain but range.
I know that the domain would be set of all real numbers.

2. That I know but I think f inverse should not exist as the function is not one one and onto?

4. Mar 25, 2015

### LCKurtz

You have a square root to worry about, so not every $x$ works in the domain. And what $x$'s work determine the range.

Sometimes when $f^{-1}$ does not exist, the notation such as $f^{-1}(2)$ means the set of all $x$ such that $f(x)=2$.

5. Mar 25, 2015

### Raghav Gupta

But I see here that if we choose any x here in this case , the value of
x2 + x + 1 is always greater than zero,
so not to worry for square root in this case.

6. Mar 25, 2015

### LCKurtz

That's right, but that does not mean the range is $(0,\infty)$. For example, what $x$ gives $f(x) = 1/2$?

7. Mar 25, 2015

### Raghav Gupta

No x gives that value

8. Mar 25, 2015

### LCKurtz

Right. So you still have to figure out the range.

9. Mar 25, 2015

### Raghav Gupta

Ya, the range then must be greater then 1/2 also.
What is the method to find the particular value?

10. Mar 25, 2015

### LCKurtz

You have to figure out the least that $x^2+x+1$, and hence its square root, can be.

11. Mar 25, 2015

### Raghav Gupta

I differentiated it,
Got 2x + 1 = 0
Hence x = -1/2, a minima.
Substituting in function we get square root of 3/4 which is √3/2
Hence range is [√3/2,∞)
Thanks.
The word least provoked the differentiation.

12. Mar 25, 2015

### LCKurtz

Good. Note that you could have also found the min value without calculus by completing the square.

13. Mar 25, 2015

### Raghav Gupta

Hmm that's also fine.
As the thread is related to relations and functions.
I wanted to ask only a last question.

If $f(x) = sin^2x + sin^2(x+ π/3) + cosxcos(x+ π/3)$ and $g(5/4)=1$, then $(gof)(x)$ is equal to?
Options are
0
1
sinx
None of these

I know gof(x) is g(f(x)) but here g(x) is not given.

14. Mar 25, 2015

### Raghav Gupta

Anybody there or should I start a new thread?

15. Mar 25, 2015

### SammyS

Staff Emeritus
I say yes - new thread. It's quite a different question.

Composition in LaTeX is $(f\circ g)(x) \ \ \ \leftarrow\ \ \ \text{(f\circ g)(x)}$ .

16. Mar 26, 2015