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The domain of a function is the set of all possible inputs (or values of x) for which the function is defined.
The given function is: \[ f(x) = 3 + (x+1)^{7/5} \]To determine its domain, we need to check the inputs for which the expression inside the function makes sense.
In this function, we have a constant term (3) and the term \[ (x+1)^{7/5} \]which means we must examine the inner expression \[ (x+1) \].
Any real number (positive, negative, or zero) can be added to 1, and raising any real number to the power of \[ \frac{7}{5} \]will result in another real number.
Therefore, the function is defined for all real numbers.
This means that the domain of \[ f(x) \] is all real numbers, or \[ (-\infty, \infty) \].

Real numbers are all the numbers we usually use and encounter in everyday life. They include:

• Integers: ...,-3, -2, -1, 0, 1, 2, 3, ...
• Rational Numbers: Numbers that can be expressed as fractions \[ \frac{a}{b} \] where a and b are integers and b is not zero.
• Irrational Numbers: Numbers that cannot be expressed as simple fractions, such as \( \pi \) and \( \sqrt{2} \).

In the context of the given problem, both the constant number 3 and the term \[ (x+1) \] raised to \[ \frac{7}{5} \] are real numbers.
This is why the function \[ f(x) = 3 + (x+1)^{7/5} \] is defined for all real numbers as long as x is a real number.

Simplifying expressions means rewriting them to make them easier to understand or work with. However, the expression must retain the original value.
The given function is:\[ f(x) = 3 + (x+1)^{7/5} \]
Here, simplifying involves checking if we can further reduce or change the form while keeping its value the same.
Let's look at the components:

• The constant term: 3
• The term involving \[ (x+1) \], raised to \[ \frac{7}{5} \].

In this case, there aren't any simpler equivalent forms for \[ (x+1)^{7/5} \].
So, \[ f(x) = 3 + (x+1)^{7/5} \] is already in its simplest form.
Simplifying expressions helps in solving equations, graphing, and other algebraic processes.