1. ## Functions

1) Simplify function
g(g−1(2x−3))

2)
State the intervals on which the following function is negative: function

f(x)= (1-x)^3(1+x)
x^6(-1-2x)

Express your answers using interval notation. For an interval (ab) , simply write (a,b). For in interval (−b) or (a)

2. The first is a definition check.

By Definition, the answer is '2x-3'. There are no calculations required. You tell me why.

3. Hi
Originally Posted by lemontea
1) Simplify function
g(g−1(2x−3))
What's the matter with this question ?

2)
State the intervals on which the following function is negative: function

f(x)= (1-x)^3(1+x)
x^6(-1-2x)
$\displaystyle f(x)=\frac{(1-x)^3(1+x)}{x^6(-1-2x)}$

You should first study the sign of $\displaystyle x\mapsto(1-x)^3$, of $\displaystyle x\mapsto1+x$, of $\displaystyle x\mapsto x,$ and of $\displaystyle x\mapsto-1-2x$ then you'll be able to conclude using the rules concerning the quotient and the product of real numbers.

4. i still dont get the second question ><''

5. Originally Posted by lemontea
i still dont get the second question ><''
If you were asked to find where is $\displaystyle x\mapsto (x-2)(x-3)$ negative, you could do this :

$\displaystyle x-3<0 \Leftrightarrow x<3$ and $\displaystyle x-2<0 \Leftrightarrow x<2$

Hence
• if $\displaystyle x<2$ then $\displaystyle \,x-3<0$ and $\displaystyle x-2<0$ hence $\displaystyle (x-3)(x-2)>0$
• if $\displaystyle 2<x<3$ then $\displaystyle \,x-3<0$ and $\displaystyle x-2>0$ hence $\displaystyle (x-3)(x-2)<0$
• if $\displaystyle x>3$ then $\displaystyle \,x-3>0$ and $\displaystyle x-2>0$ hence $\displaystyle (x-3)(x-2)>0$
We can conclude that $\displaystyle (x-3)(x-2)<0\Leftrightarrow x\in]2,\,3[$.

Is it clearer ?

6. i got the answers (-inf,-1) and (-1/2,1)...but it's incorrect..does anyone know why?

7. Originally Posted by lemontea
i got the answers (-inf,-1) and (-1/2,1)...but it's incorrect..does anyone know why?
Who says that they are incorrect? Those are the intervals that I find.