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".


The given piecewise function is defined as follows:

\[
f(x) = \begin{cases} 
\frac{3}{4}x + 1 & \text{if } x < -6 \\
\frac{x^2 - x - 6}{x} & \text{if } x \geq -6
\end{cases}
\]

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.

Consider the following piecewise-defined function. f(x) = { (3/4)x + 1 if x < -6, (x^2 - x - 6)/x if x >= -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'.

Learn how to evaluate a piecewise-defined function with conditions: (3/4)x + 1 for x < -6 and (x^2 - x - 6)/x for x >= -6. The solution focuses on evaluating the function at x = -5, covering algebraic manipulation and simplification. Ideal for students in Grades 9-10.

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