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**Exercise 1.8.** Let \( f: \mathbb{R} \to \mathbb{R} \) be a one-to-one and onto (bijective) function. Define \( g: \mathbb{R} \to \mathbb{R} \) by \( g(x) := a f(x) + b \), with \( a \neq 0 \) and \( b \) a constant. Prove that \( g \) is also a one-to-one and onto function. Then, find the formula for \( g^{-1}(x) \) (the inverse of \( g \)). Apply the given formula when \( f(x) = \frac{4x}{x-2} \).

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### Step-by-step solution:

1. **Proving that \( g \) is one-to-one (injective):**

   We need to show that for \( g(x_1) = g(x_2) \), it must follow that \( x_1 = x_2 \).
   
   \[
   g(x_1) = a f(x_1) + b, \quad g(x_2) = a f(x_2) + b
   \]
   
   If \( g(x_1) = g(x_2) \), we get:
   \[
   a f(x_1) + b = a f(x_2) + b
   \]
   Subtract \( b \) from both sides:
   \[
   a f(x_1) = a f(x_2)
   \]
   Since \( a \neq 0 \), divide by \( a \):
   \[
   f(x_1) = f(x_2)
   \]
   Because \( f \) is injective (one-to-one), this implies \( x_1 = x_2 \). Thus, \( g \) is injective.

2. **Proving that \( g \) is onto (surjective):**

   We need to show that for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( g(x) = y \).
   
   Let \( y = g(x) = a f(x) + b \). Solving for \( f(x) \):
   \[
   y - b = a f(x)
   \]
   Divide by \( a \):
   \[
   f(x) = \frac{y - b}{a}
   \]
   Since \( f \) is surjective (onto), there exists an \( x \in \mathbb{R} \) such that \( f(x) = \frac{y - b}{a} \). Therefore, \( g(x) = y \), meaning \( g \) is surjective.

   Combining injectivity and surjectivity, we conclude that \( g \) is bijective.

3. **Finding the inverse \( g^{-1}(x) \):**

   To find \( g^{-1}(x) \), we start with the equation:
   \[
   y = g(x) = a f(x) + b
   \]
   Sol

Misalkan f: R → R fungsi satu-satu dan kepada. Definisikan g: R → R dengan g(x) := a f(x) + b dengan a ≠ 0 dan b konstanta. Buktikan g juga satu-satu dan kepada, kemudian tentukan formula untuk g^{-1}(x). Terapkan formula yang ada jika f(x) := 4x / (x - 2).

This problem involves proving that the function g(x) = a f(x) + b is bijective and finding its inverse, applying it to the specific case where f(x) = 4x / (x - 2). Explore key mathematical concepts such as injective and surjective functions and linear transformations.

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