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Function evaluation means finding the value of a function for a particular input. For instance, given the functions in the exercise:

• f(x) = 30 + 5x
• g(x) = 3(1.6)^x

You can find the value of each function at specific points by substituting the values of x. For example, to find f(0) and g(0):

• f(0) = 30 + 5(0) = 30
• g(0) = 3(1.6)^0 = 3(1) = 3

This process is repeated for other points such as x = 6, x = 7, and so on. After calculating these, you compare the results to determine which function's value is higher or lower at those points.

Graph comparison involves analyzing the visual representations of functions to understand their behaviors and relationships. To compare f(x) = 30 + 5x and g(x) = 3(1.6)^x:
1. Plot both functions on the same coordinate grid.
2. Observe their intersection points, slopes, and how they grow as x increases or decreases.
For example, you'll notice that f(x) is a straight line due to it being a linear function, while g(x) curves upward or downward as it’s an exponential function. Comparing these graphs can help understand where one function surpasses the other or falls below.

The behavior of functions as x approaches infinity (either positively or negatively) determines how these functions grow or diminish over long-term. For the function f(x) = 30 + 5x:

• As x → +∞, f(x) increases linearly without bounds.
• As x → -∞, f(x) decreases linearly without bounds.

For the function g(x) = 3(1.6)^x:

• As x → +∞, g(x) increases exponentially without bounds because the base (1.6) is greater than 1.
• As x → -∞, g(x) approaches 0 because any positive number raised to a very large negative power yields a very small positive number.

Understanding these behaviors helps predict long-term trends and overall growth or decay of the functions.

Plotting on a coordinate grid is essential for visually interpreting functions. A coordinate grid has two axes: the horizontal x-axis and the vertical y-axis. When plotting f(x) = 30 + 5x and g(x) = 3(1.6)^x:
1. Choose a set of x values (both positive and negative).
2. Calculate the corresponding y values using the functions.
3. Plot these points (x, y) for both functions on the same grid.
4. Connect the dots to visualize the trend of each function.
The linear function f(x) will form a straight line, whereas the exponential function g(x) will form a curve that grows rapidly or declines, depending on the input values. Grasping these plots allows for a clearer comparison and understanding of the functions' behaviors and intersections.