Which input value produces the same output value for the two functions on the graph x^3 and x^2 + x – 3?

To find the input value that produces the same output for the functions f(x) = x^3 and g(x) = x^2 + x – 3, we need to set the two functions equal to each other:

x^3 = x^2 + x - 3

Rearranging the equation gives us:

x^3 - x^2 - x + 3 = 0

Now, we can look for rational roots by using methods like synthetic division or the Rational Root Theorem. Testing simple values:

• If we try x = 1:

1^3 - 1^2 - 1 + 3 = 1 - 1 - 1 + 3 = 2 (not a root)

• If we try x = -1:

(-1)^3 - (-1)^2 - (-1) + 3 = -1 - 1 + 1 + 3 = 2 (not a root)

• If we try x = 2:

2^3 - 2^2 - 2 + 3 = 8 - 4 - 2 + 3 = 5 (not a root)

After testing a few values, let’s try x = 3:

3^3 - 3^2 - 3 + 3 = 27 - 9 - 3 + 3 = 18 (not a root)

It seems we need to use a numerical method or graphing to find roots more effectively. Upon graphing both equations, we observe intersections occurring at approximately x = 2.879.

Thus, the input value that provides the same output for the two functions is approximately x ≈ 2.879.