Question

Graph the functions on the same coordinate plane.

f(x)=−2x

g(x)=x2−3



What are the solutions to the equation f(x)=g(x)?

Select each correct answer.

−3
−2
0
1
3

Answer 1

The correct options are,

-3, 1

Explanation:

Given functions,

`f(x)=-2x`

`g(x)=x^2-3`

`f(x)=g(x)`

`\implies -2x=x^2-3`

`\implies x^2+2x-3=0`

By middle term splitting,

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

`x(x+3)-1(x+3)=0`

`(x-1)(x+3)=0`

By zero product property,

x = 1 or x = -3

Hence, the required solutions are,

x = 1 and x = -3

Answer 2

A graph shows the solutions to be
.. x = -3
.. x = 1

Here's an analytic solution.
.. f(x) = g(x)
.. -2x = x^2 -3 . . . . . . substitute the given function definitions
.. 0 = x^2 +2x -3 . . . . add 2x
.. 0 = (x -1)(x +3) . . . . . factor

Use the zero-product rule to find values of x that are solutions. One or the other of the factors must be zero for the product to be zero.
.. x -1 = 0 . . . . first factor is zero
.. x = 1

.. x +3 = 0 . . . second factor is zero
.. x = -3