Problem 28 - The Calculus with Analytic Geometry

Question

\(f(x)=3+(x+1)^{7 / 5}\)

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty, \infty) \).

Step 1: Identify the components of the function

The given function is \( f(x) = 3 + (x+1)^{7/5} \). This function consists of a constant term, 3, and a term involving the power of \( (x+1) \) raised to the fraction \( 7/5 \).

Step 2: Determine the domain of the function

To find the domain of the function \( f(x) \), we need to ensure that the expression under the power is defined for all real numbers. Since \( (x+1) \) can be any real number, and raising a real number to the power of \( 7/5 \) is defined for all real numbers, the domain of \( f(x) \) is all real numbers, which is \( (-\infty, \infty) \).

Step 3: Simplify the expression

Rewrite the expression in a simpler form if possible. In this case, the given function \( f(x) = 3 + (x+1)^{7/5} \) is already in its simplest form.

Key Concepts

Domain of a Function

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

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

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.