Inverse Functions: Understanding the Relationship Between f⁻¹(x) and g⁻¹(x)

What is the relationship between f⁻¹(x) = -1/5x and f⁻¹(x) = 1/5x?

Is there a specific connection between these two inverse functions?

Answer:

By comparing the two inverse functions f⁻¹(x) = -1/5x and f⁻¹(x) = 1/5x, we can see that f⁻¹(x) = -g⁻¹(x).

When we reflect f⁻¹(x) = -1/5x across the x-axis, it transforms into f⁻¹(x) = 1/5x. This means that the relationship between these inverse functions is that they are essentially reflections of each other across the x-axis.

Both functions are inverse functions, but they hold different values. If we rewrite the second function as g⁻¹(x) = 1/5x, we can establish that f⁻¹(x) = -g⁻¹(x).

Therefore, f⁻¹(x) = -1/5x is reflected across the x-axis to form f⁻¹(x) = 1/5x. This relationship highlights the symmetry and transformation between these inverse functions.