31. Whichare inverses? (look at the table to answer) \( f(x)=\frac{1}{10} x \quad g(x)=10^{x} \quad h(x)=10 x \quad j(x)=\log _{10} x \)

To determine which functions are inverses, we need to check if applying one function after the other returns the input value. The functions \( f(x) = \frac{1}{10}x \) and \( g(x) = 10^x \) are inverses of each other, as \( g(f(x)) = 10^{\frac{1}{10}x} \), simplifying to return a direct transformation based on the exponential function. Similarly, \( h(x) = 10x \) and \( j(x) = \log_{10}x \) are not inverses because the linear multiplication doesn't revert fully to the logarithmic function's base conversion, so they don't satisfy the conditions needed for inverse functions. To visualize this, think of the relationship between exponential and logarithmic functions. They are like best friends who always bring the perfect solution to the table: if one goes up (the exponent), the other is ready to bring it back down (the logarithm), effectively balancing each other out! This helps you see how \( g(x) \) undoes \( f(x) \) and \( j(x) \) behaves differently from \( h(x) \). How handy!