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Understanding how to complete mathematical tables is a vital skill in mathematics and science. It involves filling in missing values based on a given formula or rule. In this exercise, the missing values are the outputs of an exponential function, which need to be calculated and entered into the table. The process starts by identifying the formula that relates the input values, in this case, the variable ‘x’, to the output values, represented by the function ‘h(x)’. Afterward, substitute each x-value into the formula and compute the result. The calculated values are then placed in the corresponding cells to complete the table.

Completing tables helps you identify patterns and understand the behavior of functions, which is especially important when dealing with complex relationships like exponential growth or decay. In real-world applications, such as computing compound interest or the decay of radioactive substances, mastery of table completion can be invaluable.

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In general, an exponential function can be written as\( f(x) = a \times b^{x} \), where ‘a’ is a coefficient, ‘b’ is the base, and ‘x’ is the exponent. They are powerful tools for modeling growth or decay processes such as population growth, bacterial proliferation, and radioactive decay.

These functions are unique because the rate of change is proportional to their value at any point which leads to a rapid increase or decrease. For instance, in our exercise, the function \( h(x) = 3.42(3^{-x/5}) \) is an exponential function with a base of 3. The negative sign in the exponent indicates that as ‘x’ increases, ‘h(x)’ will decrease — characteristic of exponential decay. Recognizing the properties of exponential functions enhances your ability to analyze and interpret real-world phenomena.

Function evaluation is a fundamental concept where you find the output of a function based on a specific input. Mathematically, this involves substituting the input value for the variable in the function’s formula and simplifying the expression to get the result.

For instance, to evaluate the function \( h(x) = 3.42(3^{-x/5}) \) at \( x = -3 \), you replace 'x' with '-3' and simplify to obtain the output. Repeating this process for various input values constructs a complete picture of the function's behavior over a range of values. Correctly evaluating functions enables you to make predictions, solve equations, and understand functional relationships in everything from simple physics to financial calculations. It’s a skill that reinforces your number sense and promotes a deeper grasp of mathematical relationships.