Math Problem Statement
Consider the following piecewise-defined function.
f(x)=34x+1x2−x−6ifx<−6ifx≥−6
Step 1 of 3 : Evaluate this function at x=−5 . Express your answer as an integer or simplified fraction. If the function is undefined at the given value, indicate "Undefined".
Solution
The given piecewise function is defined as follows:
\frac{3}{4}x + 1 & \text{if } x < -6 \\ \frac{x^2 - x - 6}{x} & \text{if } x \geq -6 \end{cases}$$ We need to evaluate this function at $$x = -5$$. Since $$-5 \geq -6$$, we will use the second part of the piecewise function: $$f(x) = \frac{x^2 - x - 6}{x}$$ Now, substitute $$x = -5$$: $$f(-5) = \frac{(-5)^2 - (-5) - 6}{-5}$$ First, calculate the numerator: $$(-5)^2 = 25, \quad -(-5) = 5, \quad \text{so} \quad 25 + 5 - 6 = 24$$ Now, divide the numerator by $$x = -5$$: $$f(-5) = \frac{24}{-5} = -\frac{24}{5}$$ Thus, the value of $$f(-5)$$ is $$-\frac{24}{5}$$. Would you like any further details or clarification on this? Here are 5 related questions: 1. How do we evaluate the function if $$x = -6$$? 2. Can we determine the domain of this piecewise function? 3. What happens if we evaluate $$f(x)$$ for values much greater than $$-6$$? 4. How do we find the points of discontinuity for this function? 5. How does this piecewise function behave at $$x = -7$$? **Tip:** Always check the condition of each piece in a piecewise function carefully before substituting values.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Rational Expressions
Formulas
(x^2 - x - 6) / x
(3/4)x + 1
Theorems
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Suitable Grade Level
Grades 9-10