Questions: Determine whether the function is continuous on the entire real number line. Explain your reasoning. f(x)=(x^2-25)^3 The function is not continuous because the function is not a polynomial. The function is not continuous because the function is not defined at x=25. The function is continuous because the function's domain is the entire real line. The function is continuous because the function is a polynomial. The function is not continuous because the function is not defined at x= ± 5.

Determine whether the function is continuous on the entire real number line. Explain your reasoning.

f(x)=(x^2-25)^3

The function is not continuous because the function is not a polynomial.
The function is not continuous because the function is not defined at x=25.
The function is continuous because the function's domain is the entire real line.
The function is continuous because the function is a polynomial.
The function is not continuous because the function is not defined at x= ± 5.

Transcript text: 1. [-/1 Points] DETAILS MY NOTES LARAPCALC10 1.6.002. Determine whether the function is continuous on the entire real number line. Explain your reasoning. \[ f(x)=\left(x^{2}-25\right)^{3} \] The function is not continuous because the function is not a polynomial. The function is not continuous because the function is not defined at $x=25$. The function is continuous because the function's domain is the entire real line. The function is continuous because the function is a polynomial. The function is not continuous because the function is not defined at $x= \pm 5$. Submit Answer Viewing Saved Work Revert to Last Response

Solution

Solution Steps

To determine whether the function $ f(x) = (x^2 - 25)^3 $ is continuous on the entire real number line, we need to check if it is a polynomial. Polynomials are continuous everywhere on the real number line. The given function is a composition of a polynomial $ x^2 - 25 $ raised to a power, which is also a polynomial. Therefore, the function is continuous on the entire real number line.

Step 1: Identify the Type of Function

The given function is:

\[ f(x) = \left(x^2 - 25\right)^3 \]

To determine whether this function is continuous on the entire real number line, we first need to identify the type of function. The expression inside the parentheses, $x^2 - 25$, is a polynomial. Raising a polynomial to a power (in this case, cubing it) results in another polynomial.

Step 2: Determine the Continuity of Polynomials

Polynomials are continuous functions. This means that they are defined and have no breaks, jumps, or holes for all real numbers. Since $f(x)$ is a polynomial, it is continuous everywhere on the real number line.

Step 3: Evaluate the Given Options

Let's evaluate the given options to determine which one correctly describes the continuity of the function:

Option 1: The function is not continuous because the function is not a polynomial.
- This is incorrect because $f(x)$ is a polynomial.
Option 2: The function is not continuous because the function is not defined at $x=25$.
- This is incorrect because $f(x)$ is defined for all real numbers, including $x=25$.
Option 3: The function is continuous because the function's domain is the entire real line.
- This is correct because the domain of a polynomial is all real numbers, and polynomials are continuous everywhere.
Option 4: The function is continuous because the function is a polynomial.
- This is also correct because polynomials are continuous on the entire real number line.
Option 5: The function is not continuous because the function is not defined at $x= \pm 5$.
- This is incorrect because $f(x)$ is defined for all real numbers, including $x = \pm 5$.