Math Problem Statement

Let g(x) be a piecewise function: g(x) = -3x^2 + 8x if x < 2, g(x) = a if x = 2, g(x) = 7x + 9 if x > 2. Determine the value of a for which g(x) is continuous at x = 2.

Solution

Let's work through the problem you've provided in the image. The piecewise function is given as:

\begin{cases} -3x^2 + 8x & \text{if } x < 2 \\ a & \text{if } x = 2 \\ 7x + 9 & \text{if } x > 2 \end{cases}$$ The question is asking about the continuity of $$g(x)$$ at $$x = 2$$. To ensure continuity, we need the left-hand limit, the function value, and the right-hand limit at $$x = 2$$ to be equal. This means: $$\lim_{x \to 2^-} g(x) = g(2) = \lim_{x \to 2^+} g(x)$$ ### Part (a): Left-hand limit as $$x \to 2^-$$ The expression for $$g(x)$$ when $$x < 2$$ is $$-3x^2 + 8x$$. We calculate the left-hand limit: $$\lim_{x \to 2^-} g(x) = -3(2)^2 + 8(2) = -3(4) + 16 = -12 + 16 = 4$$ ### Part (b): Right-hand limit as $$x \to 2^+$$ The expression for $$g(x)$$ when $$x > 2$$ is $$7x + 9$$. We calculate the right-hand limit: $$\lim_{x \to 2^+} g(x) = 7(2) + 9 = 14 + 9 = 23$$ ### Part (c): Value of $$g(2)$$ To ensure continuity, we need: $$g(2) = a$$ ### Condition for Continuity For $$g(x)$$ to be continuous at $$x = 2$$, the left-hand limit, right-hand limit, and the value of $$g(2)$$ must all be equal. This gives us the condition: $$4 = a = 23$$ Since this condition is impossible (because $$4 \neq 23$$), there is **no value of $$a$$** that makes the function continuous at $$x = 2$$. Would you like more details or have any questions on this? Here are 5 related questions that could further your understanding: 1. What does it mean for a function to be continuous at a point? 2. How do we evaluate limits from the left and right at a given point? 3. Can a function be continuous even if its piecewise parts are defined differently? 4. What would happen if the left-hand and right-hand limits were equal but the value at the point differed? 5. How does the behavior of quadratic and linear functions influence continuity at a point? **Tip**: Always check both one-sided limits when determining the continuity of a piecewise function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Piecewise Functions

Formulas

Left-hand limit: lim(x -> 2-) g(x) = -3(2)^2 + 8(2) = 4
Right-hand limit: lim(x -> 2+) g(x) = 7(2) + 9 = 23

Theorems

Continuity theorem: lim(x -> 2-) g(x) = g(2) = lim(x -> 2+) g(x)

Suitable Grade Level

Grades 10-12

Related Recommendation

Determining Continuity of a Piecewise Function at x = 2

Finding Continuity for a Piecewise Function at x = 0

Finding a Value of a for Continuity in a Piecewise Function

Study the Continuity of Piecewise Functions f(x) and g(x) with Limits

Determine the Value of 'a' for Continuity in a Piecewise Function