2x-1 berikut: a. f^-1(12) b. g^-1(15) h^-1(sqrt (7)) 3. Tentukan f^-1(x),g^-1(x),(fcirc g)^-1(x) dan (ggf)^-1(x) jika diketahui : f(x)=5-2x dan g(x)=(x-3)/(x)

Question

Pertanyaan

2x-1 berikut: a. f^-1(12) b. g^-1(15) h^-1(sqrt (7)) 3. Tentukan f^-1(x),g^-1(x),(fcirc g)^-1(x) dan (ggf)^-1(x) jika diketahui : f(x)=5-2x dan g(x)=(x-3)/(x)

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Answer 1

Let's solve these problems step-by-step. I'm assuming there's a typo in the initial expression "2x-1" and that it's not directly related to the following questions about functions f(x) and g(x).

1. Finding Inverse Function Values:

This section seems incomplete. We need the definitions of functions f(x), g(x), and h(x) to calculate their inverse function values at the specified points. Please provide the complete definitions of f(x), g(x), and h(x).

2. Finding Inverse Functions and Compositions:

Given:

* f(x) = 5 - 2x
* g(x) = (x - 3) / x

a) Finding the inverse of f(x):

1. Replace f(x) with y: y = 5 - 2x
2. Swap x and y: x = 5 - 2y
3. Solve for y: 2y = 5 - x => y = (5 - x) / 2
4. Replace y with f⁻¹(x): f⁻¹(x) = (5 - x) / 2

b) Finding the inverse of g(x):

1. Replace g(x) with y: y = (x - 3) / x
2. Swap x and y: x = (y - 3) / y
3. Solve for y: xy = y - 3 => xy - y = -3 => y(x - 1) = -3 => y = -3 / (x - 1)
4. Replace y with g⁻¹(x): g⁻¹(x) = -3 / (x - 1)

c) Finding the inverse of (f ∘ g)(x):

First, let's find (f ∘ g)(x):

(f ∘ g)(x) = f(g(x)) = f((x - 3) / x) = 5 - 2((x - 3) / x) = 5 - (2x - 6) / x = (5x - 2x + 6) / x = (3x + 6) / x

Now, find the inverse:

1. Replace (f ∘ g)(x) with y: y = (3x + 6) / x
2. Swap x and y: x = (3y + 6) / y
3. Solve for y: xy = 3y + 6 => xy - 3y = 6 => y(x - 3) = 6 => y = 6 / (x - 3)
4. Replace y with (f ∘ g)⁻¹(x): (f ∘ g)⁻¹(x) = 6 / (x - 3)

d) Finding the inverse of (g ∘ f)(x):

First, let's find (g ∘ f)(x):

(g ∘ f)(x) = g(f(x)) = g(5 - 2x) = ((5 - 2x) - 3) / (5 - 2x) = (2 - 2x) / (5 - 2x)

Now, find the inverse (this is more complex and requires careful algebraic manipulation):

1. Replace (g ∘ f)(x) with y: y = (2 - 2x) / (5 - 2x)
2. Swap x and y: x = (2 - 2y) / (5 - 2y)
3. Solve for y: x(5 - 2y) = 2 - 2y => 5x - 2xy = 2 - 2y => 5x - 2 = 2xy - 2y => 5x - 2 = y(2x - 2) => y = (5x - 2) / (2x - 2)
4. Replace y with (g ∘ f)⁻¹(x): (g ∘ f)⁻¹(x) = (5x - 2) / (2x - 2)

In summary:

* f⁻¹(x) = (5 - x) / 2
* g⁻¹(x) = -3 / (x - 1)
* (f ∘ g)⁻¹(x) = 6 / (x - 3)
* (g ∘ f)⁻¹(x) = (5x - 2) / (2x - 2)

Remember to always check your answers by verifying that f(f⁻¹(x)) = x, g(g⁻¹(x)) = x, etc. This confirms the correctness of your inverse functions.

Pertanyaan

2x-1 berikut: a. f^-1(12) b. g^-1(15) h^-1(sqrt (7)) 3. Tentukan f^-1(x),g^-1(x),(fcirc g)^-1(x) dan (ggf)^-1(x) jika diketahui : f(x)=5-2x dan g(x)=(x-3)/(x)

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