Question

NO LINKS!!! URGENT HELP PLEASE!!!

State if the given functions are inverses.

1. g(x) = 4 + (7/2)x
f(x) = 5 - (4/5)x


Find the inverses of each function.

2. g(n) = (8/3)n + 7/3

3. g(x) = 1 - 2x^3

Answer 1

1.

If the composition of two functions is the identity function, then the two functions are inverses. In other words, if f(g(x)) = x and g(f(x)) = x, then f and g are inverses.

For
`\bold{g(x) = 4 + (7)/(2)x\: and \:f(x) = 5 -(4)/(5)x}`, we have:

`f(g(x)) = 5 - (4)/(5)(4 + (7)/(2)x)\\ =5 - (4)/(5)((8+7x)/(2))\\=5 - (2)/(5)(8+7x)\\=(25-16-14x)/(5)\\=(9-14x)/(5)`

`g(f(x)) = 4 + ((7)/(5))(5 - (4)/(5)x) \\=4 + ((7)/(5))((25-4x)/(5))\\=4+ (175-28x)/(25)\\=(100+175-28x)/(25)\\=(175-28x)/(25)`

As you can see, f(g(x)) does not equal x, and g(f(x)) does not equal x. Therefore, g(x) and f(x) are not inverses.

Sure, here are the inverses of the functions you provided:

2. g(n) = (8/3)n + 7/3

we can swap the roles of x and y and solve for y to find the inverse of g(n). In other words, we can write the equation as y = (8/3)n + 7/3 and solve for n.

y = (8/3)n + 7/3

n =3/8*( y-7/3)

Therefore, the inverse of g(n) is:

`g^(-1)(n) = (3)/(8)(n - (7)/(3))=(3)/(8)*(3n-7)/(3)=\boxed{(3n-7)/(8)}`

3. g(x) = 1 - 2x^3

We can use the method of substitution to find the inverse of g(x). We can substitute y for g(x) and solve for x.

`y = 1 - 2x^3\\2x^3 = 1 - y\\x = \sqrt[3]{(1 - y)/(2)}`

Therefore, the inverse of g(x) is:

`g^(-1)(x) =\boxed{ \sqrt[3]{(1 - x)/(2)}}`

Answer 2

1) The functions are not inverses.

`\textsf{2)} \quad g^(-1)(n)=&(3)/(8)n-(7)/(8)`

`\textsf{3)} \quad g^(-1)(x)&=\sqrt[3]{(1)/(2)-(1)/(2)x}`

Explanation:

Question 1

The inverse composition rule states that if two functions are inverses of each other, then their compositions result in the identity function.

Given functions:

`g(x) = 4 + (7)/(2)x \qquad \qquad f(x) = 5 - (4)/(5)x`

Find g(f(x)) and f(g(x)):

`\begin{aligned} g(f(x))&=4+(7)/(2)f(x)\\\\&=4+(7)/(2)\left(5 - (4)/(5)x\right)\\\\&=4+(35)/(2)-(14)/(5)x\\\\&=(43)/(2)-(14)/(5)x\\\\\end{aligned}`
`\begin{aligned} f(g(x))&=5 - (4)/(5)g(x)\\\\&=5 - (4)/(5)\left(4 + (7)/(2)x \right)\\\\&=5-(16)/(5)-(14)/(5)x\\\\&=(9)/(5)-(14)/(5)x\end{aligned}`

As g(f(x)) or f(g(x)) is not equal to x, then f and g cannot be inverses.

`\hrulefill`

Question 2

To find the inverse of a function, swap the dependent and independent variables, and solve for the new dependent variable.

Calculate the inverse of g(n):

`\begin{aligned}y &= (8)/(3)n + (7)/(3)\\\\n &= (8)/(3)y + (7)/(3)\\\\3n &= 8y + 7\\\\3n-7 &= 8y\\\\y&=(3)/(8)n-(7)/(8)\\\\g^(-1)(n)&=(3)/(8)n-(7)/(8)\end{aligned}`

Calculate the inverse of g(x):

`\begin{aligned}y &= 1-2x^3\\\\x &= 1-2y^3\\\\x -1&=-2y^3\\\\2y^3&=1-x\\\\y^3&=(1)/(2)-(1)/(2)x\\\\y&=\sqrt[3]{(1)/(2)-(1)/(2)x}\\\\g^(-1)(x)&=\sqrt[3]{(1)/(2)-(1)/(2)x}\\\\\end{aligned}`