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When we talk about sketching graphs, we refer to the process of creating a visual representation of a function's behavior on a coordinate plane. Sketching is not merely plotting points; it requires an understanding of the function's properties, such as intercepts, asymptotes, increasing and decreasing intervals, and end behavior.

For instance, when sketching the graph of an exponential function like f(x) = 10^x, we should recognize that as x becomes negative, the function approaches 0 but never actually reaches it, forming a horizontal asymptote at y = 0. Conversely, as x increases, the function increases dramatically. Drawing a smooth curve through the representative points gives us a good idea of the function's shape.

An exponential function is defined by an equation in which a constant base is raised to a variable exponent. A key feature of these functions is that they grow or decay at a rate proportional to their current value. The base number in an exponential function, like 10 in f(x) = 10^x, strongly influences the rate of growth or decay. This function passes through the point (0, 1) regardless of the base because any non-zero number raised to the power of zero equals one.

Exponential functions model a wide range of real-world situations, such as compound interest, population growth, and radioactive decay. Understanding the graph of an exponential function helps in visualizing these phenomena.

In contrast, logarithmic functions are the inverse of exponential functions. The logarithmic function g(x) = log x increases as x gets larger, but unlike an exponential function, it does so at a decreasing rate. Notably, the logarithmic function has a vertical asymptote at x = 0, and it's undefined for negative values of x.

This function will cross the x-axis at x = 1, since the log of 1 is always 0. As x approaches zero from the right, the g(x) decreases without bound. Logarithmic functions are useful in many fields, including acoustics, computer science, and geology, often to describe phenomena where the rate of change slows down as the variable increases.

Considering inverse functions, they are a pair of functions that undo each other. Mathematically, if you have a function f that maps x to y, then its inverse f-1 maps y back to x. The graphs of a function and its inverse are symmetric across the line y = x.

Exponential and logarithmic functions are classic examples of inverse relationships. Taking the steps further from our original solution, recognizing the symmetry can help verify the correctness of our graphs. As seen in f(x) = 10^x and g(x) = log x, one being the inverse of the other means that for every point (a, b) on the graph of the exponential function, there is a corresponding point (b, a) on the graph of the logarithmic function.