√(7x+1) - √x = 6x

Let's solve the equation step by step:

Given:
\[ \text{can}(7x + 1) - \text{can}(x) = 6x \]

First, let's clarify what "can" represents. If "can" is a function, let's denote it as \( f \). So the equation becomes:
\[ f(7x + 1) - f(x) = 6x \]

To solve this, we need to find a function \( f \) that satisfies this equation. One way to approach this is to consider the properties of the function \( f \).

Let's assume \( f \) is a linear function of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. Then:
\[ f(7x + 1) = a(7x + 1) + b = 7ax + a + b \]
\[ f(x) = ax + b \]

Substitute these into the original equation:
\[ (7ax + a + b) - (ax + b) = 6x \]

Simplify the left-hand side:
\[ 7ax + a + b - ax - b = 6x \]
\[ 6ax + a = 6x \]

For this equation to hold for all \( x \), the coefficients of \( x \) on both sides must be equal, and the constant terms must also be equal. Therefore:
\[ 6a = 6 \]
\[ a = 1 \]

And the constant term:
\[ a = 0 \]

So, \( a = 1 \).

Thus, the function \( f(x) = x \) satisfies the given equation. Therefore, the function "can" is the identity function, \( f(x) = x \).

So the solution to the equation is:
\[ \text{can}(x) = x \]

Thus:
\[ \text{can}(7x + 1) - \text{can}(x) = 6x \]
\[ (7x + 1) - x = 6x \]
\[ 7x + 1 - x = 6x \]
\[ 6x + 1 = 6x \]

This confirms that our function \( f(x) = x \) is correct.