Answer to Question #251983 in Calculus for Ishmael Rashid

Question

Question 4

Let f be the function defined by the formula,

f(x) = 1/x+1/x − 10

.

a) Determine the largest possible domain D of f.

b) Is f injective on D?

[8,5]

Question 5

Compute the f ◦ g and its range of the functions f and g below,

f(x) = (x^2 + 5x − 6)(x^2 + 5)/|2x + 3|

, and g(x) = √x + 4

[12]

Question 6

Determine the largest domain, intersection with axes, and sign of f

f(x) = log2(2 −2/x − 3)

[16]

Answer 1

4.

a)

$D=\{x\isin (-\infin,0)\lor (0,10)\lor (10,\infin)\}$

b)

an injective function is a function f that maps distinct elements to distinct elements; that is,

$f(x_1)=f(x_2)$ implies $x_1=x_2$

So, f(x) is injective.

5.

$f\circ g=\frac{(6x-2)(x+9)}{|2\sqrt{x+4}+3|}$

Range of f(x): $(-\infin,\infin)$

Range of g(x): $[0,\infin)$

for $f\circ g$ :

$x\isin [-4,\infin)$

$f\circ g(-4)=\frac{-22\cdot 5}{3}=-\frac{110}{3}$

Range of $f\circ g$ : $[-110/3,\infin)$

6.

for f(x):

$2-\frac{2}{x-3}>0\implies \frac{2x-8}{x-3}>0$

domain: $x\isin (-\infin,3)\lor (4,\infin)$

for x-intersection:

$2-\frac{2}{x-3}=1\implies x=5$

x-intersection: $(5,0)$

for y-intersection:

$log_2(2+2/3)=log_2(8/3)=3-log_23$

y-intersection: $(0,3-log_23)$

for sign of f:

$f(x)<0\implies 0<\frac{2x-8}{x-3}<1\implies |x-3|>|2x-8|$

$f(x)<0$ at $x\isin (4,5)$

$f(x)>0$ at $x\isin (-\infin,3)\lor (5,\infin)$