Answer
a) Horizontal Line Test fails
b) 1st:$ -3 < x \leq -1$
2nd: $-1 < x \leq 2$
3rd: $2 < x \leq 4$
Work Step by Step
a)
1. Understand the Requirement for Inverses:
A function has an inverse if and only if it is one-to-one.
A one-to-one function means that for every $𝑦$-value, there is exactly one corresponding $x$-value.
2. Identify Violations of the One-to-One Condition:
Consider horizontal lines that could be drawn across the graph. Notice if any of these lines intersects the graph more than once.
For example, draw an imaginary horizontal line near the peak or trough of the function. If this line crosses the graph at more than one point, then multiple $x$-values map to the same $y$-value.
3. Conclude the Reason:
Since there are horizontal lines that intersect the graph at multiple points, the function $f$ is not one-to-one over the entire domain $[−3,4]$.
Therefore, the function does not have an inverse on this entire domain.
b)
1. Understand the Goal:
To create intervals where the function is one-to-one, find segments of the graph where the function is either strictly increasing or strictly decreasing. On such segments, no horizontal line will intersect the graph more than once.
2. Identify the Intervals:
First Interval: Look at the leftmost part of the graph from $x=−3$ to the first peak. Notice that the function is increasing. This is a candidate for one of the intervals
.
Second Interval: After the first peak, the function starts to decrease until it reaches the trough at $x=2$. This decreasing section is another candidate.
Third Interval: From the trough onward, the function starts increasing again. This is the final interval.
3. State the Intervals:
First Interval: [−3,−1] — Here, the function is strictly increasing.
Second Interval: [−1,2] — Here, the function is strictly decreasing.
Third Interval: [2,4] — Here, the function is strictly increasing again.
