You can use Desmos and copy the hyperlink to your post...... [the website calculator is "clunky," IMHO !!!! ]
Here's the graph of this function : https://www.desmos.com/calculator/s3u8kl73w3
f(x) = 4- sqrt (16 + 6x - x^2)
Notice that the term inside the square root is only valid when its ≥ 0
So......solving this we have
16 + 6x - x^2 ≥ 0 factor
(-x + 8) (x + 2) ≥ 0
Note that this is only true wnen x ≥ -2 and x ≤ 8
So....the domain of the function is [-2, 8 ]
Note to find the intersection with the y axis,, set x = 0, the term inside the square root will equal 16 ... so y = 4 - √16 = 4 - 4 = 0 ........so when x = 0, then y = 0
This is confirmed by the graph.....it contains the point (0, 0)......so it intersects the origin
And the other intersection point with the x axis is just determined by setting hte function to 0 ....so we have
0 = 4 - sqrt (16 + 6x - x^2) rearrange
sqrt (16 + 6x - x^2) = 4 square both sides
16 + 6x - x^2 = 16 subtract 16 from both sides
-x^2 + 6x = 0 factor
x ( -x + 6) = 0 set both factors to 0 and we have that the intersection points with the x axis occur at x = 0 [ which we already know ] and x = 6
Is that all you neede to know???
