Consider `f:RrarrR` given by the following. Show that 'f'is invertible. Find the inverse of 'f'.
`f(x)=5x+2`

Answer

To show that the function \( f(x) = 5x + 2 \) is invertible and to find its inverse, we will follow these steps:

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

A function is one-to-one if different inputs produce different outputs. To show that \( f \) is one-to-one, we can use the horizontal line test or algebraically prove that if \( f(a) = f(b) \), then \( a = b \).

Assume:
\[ f(a) = f(b) \]
This means:
\[ 5a + 2 = 5b + 2 \]
Subtracting 2 from both sides:
\[ 5a = 5b \]
Dividing both sides by 5:
\[ a = b \]
Since \( a = b \), we conclude that \( f \) is one-to-one.

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

A function is onto if every element in the codomain has a pre-image in the domain. Since \( f(x) = 5x + 2 \) is a linear function, we can find the range.

As \( x \) takes all real values, \( f(x) \) will also take all real values. Specifically:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to +\infty \), \( f(x) \to +\infty \).

Thus, the range of \( f \) is \( \mathbb{R} \), which matches the codomain. Therefore, \( f \) is onto.

### Step 3: Conclusion on invertibility

Since \( f \) is both one-to-one and onto, we conclude that \( f \) is invertible.

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

To find the inverse function, we start with the equation:
\[ y = f(x) = 5x + 2 \]
We need to solve for \( x \) in terms of \( y \):
1. Subtract 2 from both sides:
\[ y - 2 = 5x \]
2. Divide both sides by 5:
\[ x = \frac{y - 2}{5} \]

Thus, the inverse function is:
\[ f^{-1}(y) = \frac{y - 2}{5} \]

Since we want the inverse in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{x - 2}{5} \]

### Final Answer
The inverse of the function \( f(x) = 5x + 2 \) is:
\[ f^{-1}(x) = \frac{x - 2}{5} \]

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