Math Problem Statement
Consider the piecewise-defined function:
\[
g(x) = \begin{cases}
\frac{5}{x - 2}, & x \neq 2 \\
6, & x = 2
\end{cases}
\]
Evaluate at the given values of the independent variable: g(0), g(-6), g(2).
Solution
The given piecewise function is:
\frac{5}{x - 2}, & x \neq 2 \\ 6, & x = 2 \end{cases}$$ We are tasked with evaluating the function at three specific points: $$g(0)$$, $$g(-6)$$, and $$g(2)$$. ### 1. $$g(0)$$ Since $$0 \neq 2$$, we use the rule $$g(x) = \frac{5}{x - 2}$$. $$g(0) = \frac{5}{0 - 2} = \frac{5}{-2} = -\frac{5}{2}$$ ### 2. $$g(-6)$$ Since $$-6 \neq 2$$, we again use the rule $$g(x) = \frac{5}{x - 2}$$. $$g(-6) = \frac{5}{-6 - 2} = \frac{5}{-8} = -\frac{5}{8}$$ ### 3. $$g(2)$$ For $$x = 2$$, the function is defined as $$g(2) = 6$$, as given in the piecewise definition. ### Final answers: - $$g(0) = -\frac{5}{2}$$ - $$g(-6) = -\frac{5}{8}$$ - $$g(2) = 6$$ (already given as correct in the image) Do you have any questions or want further details on this? Here are 5 related questions you might consider: 1. How would the graph of this piecewise function look near $$x = 2$$? 2. What happens to $$g(x)$$ as $$x$$ approaches 2 from the left and right? 3. How do you find the limits of the function $$g(x)$$ as $$x$$ approaches a point where the function is piecewise defined? 4. What is the significance of defining a different value for $$g(2)$$ in this piecewise function? 5. How would you evaluate the function for a complex number input, such as $$g(i)$$? **Tip:** Always check whether the input value matches any specific conditions in a piecewise function before applying the general rule.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Formulas
g(x) = \frac{5}{x - 2}
g(2) = 6
Theorems
Definition of a piecewise function
Suitable Grade Level
Grades 9-11