Question

1- Find the domain of function f(X)=ln (-2/X2-x-6) +√x2 -1

2- Find the inverse function of the function f(X)=X2 -4x +5; X €(3,4)

3- construct the tangent line to the graph of the function f(X)= 4x.√x-2 .√x which is parallel to the line y=x.

Answer 1

1.

$$f(x)=\ln(-\dfrac{2}{x^2-x-6})+\sqrt{x^2-1}$$ $$x^2-x-6<0$$ $$x^2-1\ge0$$ $$(x+2)(x-3)<0$$ $$x\le -1\ or\ x\ge1$$

Domain:$(-2, -1]\cup[1,3)$

2.

$$𝑓(𝑥)=x^2−4x+5,x\in [3, 4]$$ $$x_v=-\dfrac{-4}{2(1)}=2$$

The function $f$ increases on $(3, 4)$

$$f(3)=(3)^2−4(3)+5=2$$ $$f(4)=(4)^2−4(4)+5=5$$

Domain: $[3, 4]$

Range: $[2, 5]$

$$y=x^2-4x+5, 3\le x\le4$$

Change $x$ and $y$

$$x=y^2-4y+5, 3\le y\le 4$$

Solve for $y$

$$y^2-4y+4=x-1$$ $$(y-2)^2=x-1$$

Since $3\le y\le 4$

$$y-2=\sqrt{x-1}$$

$$y=2+\sqrt{x-1}$$

Then

$$f^{-1}(x)=2+\sqrt{x-1}$$

Domain: $[2, 5]$

Range: $[3, 4]$

3.

$$f(x)=4x\sqrt{x}-2\sqrt{x}$$

Domain: $[0, \infin)$

$$f'(x)=4(\dfrac{3}{2})\sqrt{x}-\dfrac{2}{2\sqrt{x}}=\dfrac{6x-1}{\sqrt{x}}$$ $$slope=f'(x)=\dfrac{6x-1}{\sqrt{x}}=1$$ $$6x-\sqrt{x}-1=0$$ $$(3\sqrt{x}+1)(2\sqrt{x}-1)=0$$

Since $\sqrt{x}\ge0,$ we take $2\sqrt{x}-1=0$

$$\sqrt{x}=\dfrac{1}{2}$$ $$x=\dfrac{1}{4}$$ $$f(\dfrac{1}{4})=4(\dfrac{1}{4})\sqrt{\dfrac{1}{4}}-2\sqrt{\dfrac{1}{4}}=-\dfrac{1}{2}$$

The tangent line to the graph is

$$y+\dfrac{1}{2}=x-\dfrac{1}{4}$$ $$y=x-\dfrac{3}{4}$$