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Consider the following function:

[tex]\[ k(x)=\left\{\begin{array}{ll}
\frac{2}{x} & \text{if } x\ \textless \ 1 \\
-\frac{8}{7} x & \text{if } x\ \textgreater \ 1
\end{array}\right. \][/tex]

Step 1 of 3: Identify the general shape and direction of the graph of this function on the interval \((- \infty, 1)\).

Answer:


Sagot :

To analyze the function \( k(x) \) on the interval \((-\infty, 1)\), we need to focus on the piece of the function defined for \( x < 1 \). This part of the function is given as:

[tex]\[ k(x) = \frac{2}{x} \][/tex]

1. General Shape:
- The function \( k(x) = \frac{2}{x} \) is a hyperbola. A hyperbola typically has two asymptotes and is symmetric about these asymptotes.

2. Asymptotic Behavior:
- As \( x \) approaches \( 0 \) from the left (negative side), \( \frac{2}{x} \) becomes very large negative.
- As \( x \) approaches \( 0 \) from the right (positive side), \( \frac{2}{x} \) becomes very large positive.

3. Behavior at Endpoints:
- As \( x \) approaches \(-\infty\), \( \frac{2}{x} \) approaches \( 0 \) from the negative side.
- As \( x \) approaches \( 1 \) from the left, \( \frac{2}{x} \) approaches \( 2 \) because \( \frac{2}{1} = 2 \).

4. Decreasing/Increasing Intervals:
- On the interval \((-\infty, 0)\), the function \( \frac{2}{x} \) is negative and as the value of \( x \) becomes more negative, \( \frac{2}{x} \) approaches \( 0 \), meaning the function is decreasing.
- On the interval \((0, 1)\), the function \( \frac{2}{x} \) is positive and as the value of \( x \) approaches \( 1 \), \( \frac{2}{x} \) approaches \( 2 \), meaning the function is increasing.

From this, we deduce that:

- The general shape of the function \( k(x) = \frac{2}{x} \) on the interval \((-\infty, 1)\) is a hyperbola.
- The function \( k(x) \) is decreasing on the interval \((-\infty, 0)\) and increasing on the interval \((0, 1)\).

Thus, the answer is:

- Shape: Hyperbola
- Direction: Decreasing on [tex]\((-\infty, 0)\)[/tex] and increasing on [tex]\((0, 1)\)[/tex]