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A) All real numbers

B) All integers

C) All even numbers

D) All odd numbers

**Correct answer: A) All real numbers**

The domain of the function f(x) = 5/6[3/5]x is the set of all possible values of x for which the function is defined.

**The function f(x) = 5/6[3/5]x can be simplified to f(x) = (9/25)x.**

Since this is a polynomial function with a single variable, there are no restrictions on the domain. In other words, any real number can be plugged in for x and the function will be defined.

The domain of a function is the set of all possible input values for which the function produces a meaningful output. In other words, it's the set of all x-values that can be plugged into the function to get a real number as a result.

Some functions may have restrictions on the domain. For instance, a function that involves taking the square root of a number cannot accept negative numbers as input since the square root of a negative number is not defined in the real number system.

Similarly, a function that involves dividing by zero cannot accept zero as input since division by zero is undefined. In such cases, the domain of the function needs to be restricted to exclude the values that lead to undefined results.

A function is a mathematical rule that assigns a unique output to each input. In this question, we are given a function f(x) = 5/6[3/5]x, and we are asked to determine its domain.

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of values of x for which the function f(x) produces a real number output.

**To evaluate the function f(x) = 5/6[3/5]x, we first need to simplify the expression inside the square brackets:**

[3/5]x = 3/5 * x

Now, we can substitute this expression into the original function:

f(x) = 5/6 * (3/5 * x)

Multiplying the fractions, we get:

f(x) = 1/2 * x

From the simplified expression for f(x), we can see that the function is defined for all real numbers. Therefore, the domain of f(x) is the set of all real numbers, or

Domain = (-∞, ∞)

However, in the case of f(x) = 5/6[3/5]x, we can see that there are no variables in the denominators or under square roots. The expression [3/5]x simply represents a scalar multiplication of the variable x by the constant 3/5. Multiplying a variable by a constant does not impose any restrictions on the possible input values of the function.

As a result, f(x) is defined for all real values of x, including positive and negative numbers, fractions, and irrational numbers. Therefore, the domain of f(x) is (-∞, ∞), which means that any real number can be plugged into the function without causing it to be undefined.

It's also important to understand the significance of the domain in relation to the graph of the function. The domain of a function essentially defines the set of all x-values that can be plotted on the graph, and any values of x that are not in the domain will not appear on the graph.

If we were to graph the function f(x) = 5/6[3/5]x, we would plot the function for all values of x that are in the domain (-∞, ∞). The resulting graph would be a straight line with a slope of 1/2, passing through the origin (0,0). The slope of the line indicates how much the output of the function changes for each unit change in the input, and the fact that the line passes through the origin indicates that the function has a y-intercept of 0.

Understanding the domain of a function is an important aspect of calculus and other mathematical disciplines. By determining the set of all possible input values, we can ensure that the function is well-defined and identify any features of the graph that might affect its behavior. This knowledge can be applied to a wide range of problems and applications in fields such as engineering, physics, economics, and more.

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