Question

invertible then fill in the sentence for
the best possible justification. if the
function is invertible, state the
domain and range of its inverse.

answer
attempt 1 out of 2

domain of function:

range of function:

the function
because it
.

in other words,
inputs are mapped to

output

Explanation:

Step1: Determine the domain of the function

The graph is a semicircle (upper half, since it's above the x - axis) centered at some point. Looking at the x - values, the leftmost point is at \(x=-8\) and the rightmost point is at \(x = 2\). So the domain of the function (the set of all x - values for which the function is defined) is from \(-8\) to \(2\), or in interval notation \([-8,2]\).

Step2: Determine the range of the function

The range is the set of all y - values. The lowest y - value on the graph is \(4\) (wait, no, looking at the graph, the y - axis: the bottom of the semicircle? Wait, no, the graph is a upper - half semicircle? Wait, the y - values: the minimum y - value? Wait, the graph starts at x=-8, y = 4? No, looking at the grid, the y - axis has values from 0 to 10. Wait, the graph is a semicircle with the top at y = 9, and the bottom (the endpoints) at y = 4? Wait, no, let's re - examine. The x - axis: from - 10 to 10. The graph is a curve that starts at x=-8, y = 4 (approx) and goes up to a peak at x=-3 (maybe), y = 9, then comes down to x = 2, y = 4. So the range of the function (the set of y - values) is from \(4\) to \(9\), so the range is \([4,9]\).

Step3: Check if the function is invertible

A function is invertible if and only if it is one - to - one (passes the horizontal line test). A horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one - to - one. For this semicircle (a curve that is symmetric in a way that a horizontal line will intersect it at two points: for example, a horizontal line at y = 6 will intersect the graph at two different x - values). So the function is not invertible because it fails the horizontal line test (a horizontal line intersects the graph at more than one point), meaning it is not one - to - one (multiple inputs map to the same output).

Answer:

Domain of function: \(\boldsymbol{[-8, 2]}\)
Range of function: \(\boldsymbol{[4, 9]}\)
The function \(\boldsymbol{\text{is not invertible}}\) because it \(\boldsymbol{\text{fails the horizontal line test (multiple inputs map to the same output)}}\).
In other words, \(\boldsymbol{\text{multiple}}\) inputs are mapped to \(\boldsymbol{\text{the same}}\) output.