Math Problem Statement
Solve question 8
Solve question 8: Let p be the piecewise defined function p(x) = 3 - 2g(x) for x < 3 and p(x) = 5k + f(x) for x ≥ 3. Find the value of k that makes p continuous at x = 3.
Solution
To solve question 8, we need to find the value of that makes continuous at .
The piecewise function is given as:
3 - 2g(x), & x < 3 \\ 5k + f(x), & x \geq 3 \end{cases}$$ To ensure continuity at $$x = 3$$, the left-hand limit (as $$x \to 3^-$$) must equal the right-hand limit (as $$x \to 3^+$$), and both must equal the value of the function at $$x = 3$$. ### Step 1: Evaluate the left-hand limit as $$x \to 3^-$$ For $$x < 3$$, the function is $$p(x) = 3 - 2g(x)$$. Therefore: $$\lim_{x \to 3^-} p(x) = 3 - 2g(3)$$ From the graph of \( g(x)Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit = Right-hand limit for continuity
p(x) = 3 - 2g(x) for x < 3
p(x) = 5k + f(x) for x ≥ 3
Theorems
Continuity Theorem
Limit Definition for Continuity
Suitable Grade Level
Grades 10-12