How do you use the horizontal line test to determine whether the function #f(x)=1/8(x+2)^2-1# is one to one?

Question

James Allen · Accepted Answer

The horizontal line test is to drawing several horizontal lines, #y=n,ninRR#, and see if any lines cross the function more than once. A one-to-one function is a function where each #y# value is given by only one #x# value,, whereas a many-to-one function is a function where multiple #x# values can give 1 #y# value. If a horizontal line crosses the function more than once, then it means that the function has more than one #x# value which gives one value for #y#. In this case, doing so will give two intersections for #y>1# Example:
 graph{(y-(x+2)^2/8+1)(y-1)=0 [-10, 10, -5, 5]}  The line #y=1# crosses #f(x)# twice and is not a one-to-one function.

Sadie Alexander · Answer

undefined To use the horizontal line test to determine whether the function $f(x) = \frac{1}{8}(x+2)^2 - 1$ is one-to-one, we need to check if every horizontal line intersects the graph of the function at most once. If it does, then the function is one-to-one.

The function is in the form of a parabola, which opens upwards and is symmetric about the vertical line passing through the vertex (-2, -1). Since it's a parabola, any horizontal line will intersect the graph at most once, except at the vertex where it touches the graph.

Therefore, the function $f(x) = \frac{1}{8}(x+2)^2 - 1$ is one-to-one, as every horizontal line intersects its graph at most once.