Graph f(x) = 2^x and g(x) = (1/2)^x on the same axes from x = -3 to x = 3. Show their horizontal asymptotes and intersection point.

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Let's graph two exponential functions: f of x equals 2 to the power of x, and g of x equals one-half to the power of x, on the domain from negative 3 to positive 3. The function f of x is shown in blue and increases from left to right, while g of x is shown in red and decreases. Both functions have a horizontal asymptote at y equals 0, shown by the green dashed line. The two functions intersect at the point (0, 1), marked in purple. Notice that g of x equals 2 to the power of negative x, making these functions reflections of each other across the y-axis. Let's examine the key properties of these exponential functions. First, f of x equals 2 to the x is always positive and increases as x increases. Similarly, g of x equals one-half to the x, which equals 2 to the negative x, is also always positive but decreases as x increases. Both functions have a horizontal asymptote at y equals 0. They intersect at the point (0, 1) because 2 to the power of 0 and one-half to the power of 0 both equal 1. An important relationship between these functions is that they are reflections of each other across the y-axis, as shown by the yellow arrows. This is because g of x equals 2 to the negative x, which means g of x equals f of negative x. Let's find the intersection point of these two functions algebraically. We set f of x equal to g of x, which gives us 2 to the x equals one-half to the x. Since one-half to the x equals 2 to the negative x, we can rewrite this as 2 to the x equals 2 to the negative x. For this equation to be true, x must equal negative x, which means 2x equals 0, so x equals 0. Now we can find the y-coordinate by evaluating either function at x equals 0. f of 0 equals 2 to the power of 0, which is 1. Similarly, g of 0 equals one-half to the power of 0, which is also 1. Therefore, the intersection point is (0, 1), as shown on our graph with the purple dot and dashed lines. Now let's examine the horizontal asymptotes of these functions. For f of x equals 2 to the x, as x approaches negative infinity, the function value approaches 0 from above. This is because very large negative powers of 2 result in very small positive numbers. For example, 2 to the negative 10 is approximately 0.001. The limit of 2 to the x as x approaches negative infinity is 0. Similarly, for g of x equals one-half to the x, as x approaches positive infinity, the function value approaches 0 from above. This is because very large positive powers of one-half result in very small positive numbers. The limit of one-half to the x as x approaches positive infinity is 0. Therefore, both functions have the same horizontal asymptote at y equals 0, shown by the green dashed line. To summarize what we've learned: We graphed two exponential functions, f of x equals 2 to the x and g of x equals one-half to the x, on the domain from negative 3 to positive 3. Both functions have a horizontal asymptote at y equals 0. The function f of x approaches this asymptote as x approaches negative infinity, while g of x approaches it as x approaches positive infinity. We found that these functions intersect at the point (0, 1) by solving the equation 2 to the x equals one-half to the x. An important relationship between these functions is that they are reflections of each other across the y-axis. This is because g of x equals one-half to the x can be rewritten as 2 to the negative x, which means g of x equals f of negative x. This reflection property is a general characteristic of exponential functions with reciprocal bases.