Answer to Question #251981 in Calculus for Ishmael Rashid

Question

Question 1

Solve the following equation

a) 2e^2x−1 + 5e^x2= 0

b) 2^4x + 2^2x−1 > 2

[8,6]

Question 2

Let f : (−∞, 2] → R be given by

f(x) = √2 − x

a) Show that f is injective.

b) Determine im(f).

c) Find a left inverse g : R → (−∞, 2] of f.

[7,6,5]

Question 3

Let f : Z → Z be given by

f(z) = (2z − 5, if z ≥ 0;

(z + 5, if z < 0.

a) Is f an injective function?

b) Let u ∈ Z, u ≤ 5. show that u ∈ im(f).

c) Let v ∈ Z, v > 5, Show that v ∈ im(f) if and only if v + 5 is even .

[5,6,6]

Answer 1

Solution.

Question 1

a)

equation $2e^{2x-1}+5e^{x^2}=0$ has not real roots because $a^{f(x)}>0, a>0, a\neq1$ for all x.

b)

$2^{4x}+2^{2x-1}>2$

$2•2^{4x}+2^{2x}-2•2>0$

Let be $t=2^{2x},$ then

$2t^{2}+t-4>0$

From here $t<\frac{-1-\sqrt{33}}{4},$ and $t>\frac{-1+\sqrt{33}}{4}.$

$2^{2x}<\frac{-1-\sqrt{33}}{4}$ has not real solutions.

$2^{2x}>\frac{-1+\sqrt{33}}{4},$

$2x>\log_2{\frac{-1+\sqrt{33}}{4}},$

$x>\frac{1}{2}\log_2\frac{-1+\sqrt{33}}{4}.$

Question 2

$f:(-\infty,2]\to R, f(x)=\sqrt{2-x}$

a) each x from $(-\infty,2)$ has y to correspond, for a example, f(0)=$\sqrt{2},$ f(-2)=2, f(-7)=3.

So, f(x) is injective function.

b) as $x\in (-\infty,2]$ f(x) is an increasing function in the domain

$x\to -\infty, f(x)\to +\infty$

So, Im(f)=$+\infty.$

c) find a left inverse of f:

$f^2=2-x,\newline x=2-f^2,\newline g(x)=-x^2+2, x\in [0,+\infty).$

Question 3

$f:Z\to Z, f(z)=2z-5, if z\geq0; z+5, if z<0.$

a) $f(z)=0, z=2.5, \text{or } z=-5.$

So, f(z) is not injective.

b) such as $z\leq 5,$ we have

$z\in (-\infty,5]\implies u\leq 5 \in$ im(f).

c) $v\in Z,v>5.$

Let be $v=7,$ 7+5=12, 12 is even.