Show that `f:Q to Q, f(x) = 5x+4` is a bijection and find the inverse of 'f'.

Answer

To show that the function \( f: \mathbb{Q} \to \mathbb{Q} \) defined by \( f(x) = 5x + 4 \) is a bijection and to find its inverse, we will follow these steps:

### Step 1: Show that \( f \) is injective

To prove that \( f \) is injective, we need to show that if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
2. This means \( 5x_1 + 4 = 5x_2 + 4 \).
3. Subtract 4 from both sides:
\[ 5x_1 = 5x_2 \]
4. Divide both sides by 5:
\[ x_1 = x_2 \]

Since \( x_1 = x_2 \) whenever \( f(x_1) = f(x_2) \), \( f \) is injective.

### Step 2: Show that \( f \) is surjective

To prove that \( f \) is surjective, we need to show that for every \( y \in \mathbb{Q} \), there exists an \( x \in \mathbb{Q} \) such that \( f(x) = y \).

1. Let \( y \) be any rational number.
2. We set up the equation:
\[ y = 5x + 4 \]
3. Rearranging gives:
\[ 5x = y - 4 \]
4. Dividing both sides by 5:
\[ x = \frac{y - 4}{5} \]

Since \( y \) is rational, \( y - 4 \) is also rational, and dividing a rational number by 5 gives another rational number. Thus, \( x \) is rational.

This shows that for every \( y \in \mathbb{Q} \), there exists an \( x \in \mathbb{Q} \) such that \( f(x) = y \). Therefore, \( f \) is surjective.

### Step 3: Conclusion on bijectivity

Since \( f \) is both injective and surjective, we conclude that \( f \) is bijective.

### Step 4: Find the inverse of \( f \)

To find the inverse function \( f^{-1} \), we start from the equation we derived for \( y \):

1. From \( y = 5x + 4 \), we already rearranged it to find \( x \):
\[ x = \frac{y - 4}{5} \]
2. We can denote the inverse function as:
\[ f^{-1}(y) = \frac{y - 4}{5} \]

To express the inverse function in terms of \( x \):
\[ f^{-1}(x) = \frac{x - 4}{5} \]

### Final Result

Thus, the function \( f(x) = 5x + 4 \) is bijective, and its inverse is given by:
\[ f^{-1}(x) = \frac{x - 4}{5} \]

---