Helpppp.. Algebraa.. Urgent

Question

Find the exact value of each x∈[0,2π) for which cos^2 (x)= (1/2) sin(2x).

If g(x)=5+3x/ x−3

1. What is the domain of g(x)?
2. Verify (with sketch and/or short argument) that g is a one-to-one function.
3. Find a formula for the inverse function: that is, find g−1(x).
4. Find the range of g.

Answer 1

0. cos²x = ½sin 2x = (sin x)(cos x) means that

cos²x – (sin x)(cos x) = (cos x)(cos x – sin x) = 0

So either cos x = 0, which is true only at π/2 and 3π/2

or cos x – sin x = 0 which means that

cos x = sin x

This can only true in quadrant 1 where cos x and sin x are both positive or or quadrant 3 where they are both negative (in quadrants 2 and 4 they have opposite signs).

In quadrant 1 cos x = sin x only at x = π/4

In quadrant 3 cos x = sin x only at x = 5π/4

So the four angles in increasing order are

π/4, π/2, 5π/4, 3π/2

Questions 1–3 below are about g(x) = (5 + 3x)/(x – 3)

1. What is the domain of g(x)?

all values of x except 3, so –∞ < x < 3 ∪ 3 < x < ∞

or in interval notation (–∞,3) ∪(3,∞)

2. Verify that g is a one-to-one function

Over the part of the domain –∞ < x < 3

g(x) is uniformly decreasing from a maximum value approaching (but never quite reaching) 3 on the left

and asymptotically approaches –∞ as x approaches 3.

ver the part of the domain 3 < x < ∞

(x) is uniformly decreasing from approaching positive infinity at x = 3 and approaches (but never quite reaches) a minimum value of 3

Since no two values of x produce the same values of g(x), g is a one-to-one function.

3. Find g⁻¹(x).

Take y = (5 + 3x)/(x – 3) = g(x)

and switch x and y to get

x = (5 + 3y)/(y – 3)

Now solving for y will give the formula of the inverse function y = g⁻¹(x).

The key step is using long division to find that

(5 + 3y)/(y – 3) = 3 + 14/(y – 3)

Now we have

x = 3 + 14/(y – 3) which is much easier to solve for y

Subtracting 3 from both sides gives

x – 3 = 14/(y – 3)

Now multiply both sides by (y – 3)/(x – 3) to obtain

y – 3 = 14/(x – 3)

Adding 3 to both sides gives

y = 3 + 14/(x – 3)

Substitute 3 = 3(x – 3)/(x – 3) to obtain

y = 3(x – 3)/(x – 3) + 14/(x – 3) = (3x – 9)/(x – 3) + 14/(x – 3)

where the right-hand can be written as a single fraction (3x – 9 + 14)/(x – 3)

Now we have

y = (3x – 9 + 14)/(x – 3) = (3x + 5)/(x – 3)

So our solution is

g⁻¹(x) = (3x + 5)/(x – 3)

Notice that that the inverse function g⁻¹(x) is the same as the original function g(x).

I recommend using a graphing calculator (like Desmos) and plotting

the function y = (5 + 3x)/(x – 3) and the line y = x

What you'll find is that the function is symmetrical about the y = x line.

Any function that has this symmetry will be identical with its inverse. The reciprocal function f(x) = 1/x is another example of such a function.

g(x) = (5 + 3x)/(x – 3) can be rewritten as g(x) = 14/(x – 3) + 3

which is just a version of the reciprocal function

1. shifted 3 units up and 3 units to the right, and
2. stretched by a factor of 14

4. Find the range of g.

–∞ < y < 3 ∪ 3 < y < ∞

or in interval notation (–∞,3) ∪(3,∞)

Plot g(x) on Desmos and look for the vertical and horizontal asymptotes.

Answer 2

cos^2(x) = (1/2) sin(2x)

Using the double angle formula,

sin(2x) = 2× sin(x) × cos(x)

cos^2(x) = sin(x) × cos(x)

Subtracting the right-hand side of the equation from both sides,

cos^2(x) - sin(x)×cos(x) = 0

cos(x) × ( cos(x) - sin(x)) = 0

So, either cos(x) = 0 or cos(x) = sin(x).

In the indicated set, cos(x) = 0 for x = π/2 and 3π/2.

In the indicated set, sin(x)=cos(x) for x=π/4 and 5π/4.

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I presume g(x) = 5 + 3x/(x-3).

The domain is all real numbers except x=3, because this would lead to division by zero.

Here is a graph of the function:

https://www.desmos.com/calculator/tmq4quzyra

If you perform a "horizontal line" test, you will see there no values that would prevent this function being invertible: it is one-to'-one.

To find the inverse:

First exchange the x's and the y's;

x = 5+ 3y/(y-3)

Solve for y:

x-5 = 3y/(y-3)

(x-5)×(y-3) = 3y

xy +15 -5y -3x = 3y

15 - 3x = 3y + 5y -xy

15 - 3x = 8y - xy = (8-x)y

y = (15 -3x)/(8-x) = 3×(5-x)/(8-x) = g^-1(x)

The range of g(x) is all real numbers except y=8.