Find A Constant That Makes It Continuous On The Interval

Question

find A Constantcthat Makesgcontinuous On Îíì³ 420422xcx Determine the value of a constant $ c $ that ensures the function $ g(x) $ remains continuous over its entire domain. The problem involves analyzing the behavior of the function at potential points of discontinuity and applying the definition of continuity, which requires that the limit of $ g(x) $ as $ x $ approaches a point equals the function's value at that point.

Dr. Jack HW Helper · Accepted Answer

To find the constant $ c $ that makes the function $ g(x) $ continuous on its domain, it is essential first to understand the functional form and the points where potential discontinuities could occur. Typically, the focus is on points where the functional expression changes form or is undefined, such as at the boundaries of piecewise functions or points where denominators vanish. Assuming $ g(x) $ is defined as a piecewise function involving $ c $, for instance, such as: $$ g(x) = \begin{cases} x^2 + c, & \text{if } x 2x + c, & \text{if } x \geq a \end{cases} $$ To ensure continuity at $ x = a $, the limit of $ g(x) $ as $ x $ approaches $ a $ from the left must equal the limit from the right, and both must equal $ g(a) $: $$ \lim_{x \to a^-} g(x) = g(a) = \lim_{x \to a^+} g(x) $$ Applying this principle, we set up equations: $$ \lim_{x \to a^-} (x^2 + c) = 2a + c $$ $$ a^2 + c = 2a + c $$ Solving for $ c $, the $ c $ cancels out, indicating that the value of $ c $ doesn't affect continuity at $ a $, or if the functional forms are different, similar steps are taken at each critical point. Without the explicit form of $ g(x) $, the general approach involves identifying the points of potential discontinuity, computing the limits on both sides, and equating these limits to the function's value at these points. Simplifying these equations yields the specific $ c $ that guarantees continuity. If the functional definitions involve parameters or more complex expressions, similar limits and algebraic manipulations apply. Thus, the key is understanding limits, functional behaviors, and ensuring the matching of limit values with function values at critical points. End of the Paper References Lay, D. C. (2012). Calculus and its applications. Pearson Education. Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning. Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry. Addison Wesley. Swokowski, E. W., &

find A Constantcthat Makesgcontinuous On Îíì³ 420422xcx

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