Question Statement
Consider the function:
f(x)=x3+8x
Determine whether f(x) is continuous on the following intervals:
(a) [−4,−3]
(b) [−10,10]
Background and Explanation
A rational function (ratio of polynomials) is continuous everywhere except at values of x that make the denominator equal to zero. To determine continuity on a closed interval, we must verify that no points of discontinuity fall within that interval.
Solution
First, we factor the denominator to identify where f(x) is undefined. Using the sum of cubes formula a3+b3=(a+b)(a2−ab+b2):
f(x)=(x+2)(x2−2x+4)x
The denominator equals zero when x+2=0 or x2−2x+4=0.
- x+2=0⇒x=−2
- x2−2x+4=0 has discriminant Δ=(−2)2−4(1)(4)=4−16=−12<0, so no real solutions.
Therefore, f(x) is undefined only at x=−2, making this the only point of discontinuity.
We check whether the discontinuity at x=−2 lies within the interval [−4,−3].
Since −3<−2 (that is, −2 is to the right of −3 on the number line), we have:
−2∈/[−4,−3]
Because f(x) is defined for all x∈[−4,−3] and rational functions are continuous on their domain, f(x) is continuous on the interval [−4,−3].
We check whether x=−2 lies within [−10,10].
Since −10≤−2≤10, we have:
−2∈[−10,10]
At x=−2, the function f(x) is undefined (division by zero occurs). Therefore, f(x) is discontinuous on the interval [−10,10] due to the infinite discontinuity at x=−2.
- Sum of cubes factorization: a3+b3=(a+b)(a2−ab+b2)
- Continuity of rational functions: Continuous everywhere except where denominator equals zero
- Quadratic discriminant: Δ=b2−4ac to determine if quadratic factors have real roots
- Interval membership testing: Verifying whether critical points fall within closed intervals [a,b]
Summary of Steps
- Factor the denominator x3+8 using sum of cubes to get (x+2)(x2−2x+4)
- Find zeros of denominator: Solve x+2=0 to get x=−2 (the only real discontinuity)
- For part (a): Verify that −2∈/[−4,−3], concluding f is continuous on [−4,−3]
- For part (b): Verify that −2∈[−10,10], concluding f is discontinuous on [−10,10]