Problem 25 - Calculus: Early Transcendentals

Question

Continuity Determine the interval(s) on which the following functions are continuous. $$\text { 5. } p(x)=4 x^{5}-3 x^{2}+1$$

Step-by-Step Solution

Answer

Answer: \((-\infty, \infty)\)

Step 1: Check for Undefined Points

Observe the function: $$ p(x)=4x^{5}-3x^{2}+1 $$ Since this is a polynomial function, there are no denominators, square roots, or logarithms that might cause the function to be undefined. Therefore, there are no undefined points in the function.

Step 2: Check for Jumps or Discontinuities

As we mentioned before, polynomial functions have no jumps or discontinuities; they are continuous on their entire domain. Thus, for the given polynomial function \(p(x)=4x^{5}-3x^{2}+1\), there are no jumps or discontinuities.

Step 3: Determine the Interval(s) of Continuity

Since there are no undefined points, jumps, or discontinuities in the polynomial function, we can conclude that it is continuous everywhere. In interval notation, this means that the function is continuous on the interval \((-\infty, \infty)\). In conclusion, the given function \(p(x)=4x^{5}-3x^{2}+1\) is continuous on the interval \((-\infty, \infty)\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. Examples include expressions like \(4x^5 - 3x^2 + 1\). Key characteristics of polynomial functions include:

• They can have multiple terms, where the degree (the highest exponent) determines the function's overall behavior.
• They do not have variables in the denominator, which means they are not fractions or radical expressions.
• Polynomial functions are smooth and continuous across their domain.

These features simplify analyzing polynomial functions, as they avoid potential problems like undefined points. For example, in our case \(p(x)=4x^5 - 3x^2 + 1\), the terms only involve positive integer powers of \(x\), making it a simple polynomial with full continuity.

Intervals of Continuity

Intervals of continuity are the ranges over which a function maintains continuous behavior. For a polynomial function like \(p(x)=4x^5 - 3x^2 + 1\), continuity is straightforward. Since polynomial functions are inherently continuous for all real numbers, their intervals of continuity usually span from negative infinity to positive infinity,

• Polynomial functions don't face issues like division by zero or complex numbers when \(x\) is any real number.
• This absence of potential "problem" points ensures seamless behavior throughout.

Therefore, the interval of continuity for the given polynomial function is \((-\infty, \infty)\). This implies that there is no break or pause in the graph of the function across these intervals.

Discontinuities

Discontinuities in a function refer to points or intervals where a function is not continuous. They can occur for several reasons, such as having terms like division by zero, square roots of negative numbers, or jumps in piecewise functions. However, polynomial functions like \(4x^5 - 3x^2 + 1\) do not exhibit these characteristics:

• They avoid division by zero since they're not fractions.
• There are no square roots that could lead to undefined conditions.
• Polynomial graphs are smooth, without sudden jumps or breaks.

In our given function, there are absolutely no points of discontinuity. Hence, \(p(x)=4x^5 - 3x^2 + 1\) is perfectly continuous across the entire set of real numbers, meaning \((-\infty, \infty)\) is free from any disruptions.