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In mathematics, an inverse function is a function that 'reverses' another function. It essentially undoes the action of the original function. For any function \(f\), the inverse function, denoted \(f^{-1}\), swaps the roles of the input and the output.

To find the inverse of a function, follow these steps:

• Rewrite the function replacing \(f(x)\) with \(y\).
• Solve the equation for \(x\).
• Lastly, swap \(x\) and \(y\) and replace \(y\) with \(f^{-1}(x)\) to get the final inverse function.

For example, let's find the inverse of \(g(x) = -6x - 8\).

• First, replace \(g(x)\) with \(y\): \(y = -6x - 8\).
• Solve for \(x\):
  • Add 8 to both sides: \(y + 8 = -6x\).
  • Divide both sides by -6: \(x = - \frac{y + 8}{6} \).
• Finally, replace \(y\) with \(g^{-1}(x)\): \(g^{-1}(x) = - \frac{x + 8}{6} \).

The inverse is obtained by reflecting the function over the line \(y = x\), making it a crucial concept in understanding the relationship between functions and their inverses.

A function is strictly decreasing if, as the input values increase, the output values always decrease. This means that for any two values \(x_1\) and \(x_2\) where \(x_1 < x_2\), the function \(f\) satisfies \(f(x_1) > f(x_2)\).

Let's look at the function \(g(x) = -6x - 8\). The coefficient of \(x\) is -6, which is negative. A negative coefficient indicates that the function is decreasing. Therefore, the function continues to decrease as \(x\) increases.

Verifying if a function is strictly decreasing helps in determining if it is one-to-one. A strictly decreasing function passes the horizontal line test and is therefore a one-to-one function. This concept simplifies various mathematical analyses and problem-solving scenarios.

For example: if \(x_1 = 1\) and \(x_2 = 2\), then for \(g(x) = -6x - 8\), we find

• \(g(1) = -6(1) - 8 = -6 - 8 = -14\)
• \(g(2) = -6(2) - 8 = -12 - 8 = -20\).

Notice how \(g(1) > g(2)\), confirming that \(g(x)\) is strictly decreasing.

The horizontal line test is a method used to determine if a function is one-to-one. If every horizontal line intersects the graph of the function at most once, then the function is one-to-one.

For the function \(g(x) = -6x - 8\), since this function is strictly decreasing (as shown earlier), it will pass the horizontal line test.

• Draw a horizontal line; if it intersects the graph more than once, the function is not one-to-one.
• For \(g(x) = -6x - 8\), no horizontal line will intersect the graph more than once, confirming it's a one-to-one function.

Ensuring a function is one-to-one before finding its inverse is crucial because only one-to-one functions have inverses that are also functions. Understanding this concept is essential for solving problems related to functions and their inverses efficiently.