Question

Two functions are combined resulting in the function j(x) = -3x^2+5.

Which operations applied to the specified functions could result in j(x)? Select three options.

-two linear functions by addition
-two quadratic functions by addition
-a linear function and a quadratic function by addition
-two linear functions by multiplication
-two quadratic functions by multiplication

Answer 1

Explanation:

In this exercise, we have the following function that result from a combination of two functions:

$j(x) = -3x^2+5$

So the operations applies that could result in $j(x)$ are:

1. Two quadratic functions by addition:

For instance, it could be:

$f(x)=-4x^2+3 \\ \\ g(x)=x^2+2 \\ \\ \\ j(x)=f(x)+g(x) \\ \\ j(x)=(-4x^2+3)+(x^2+2) \\ \\ j(x)=(-4x^2+x^2)+(3+2) \\ \\ j(x)=-3x^2+5$

2 linear function and a quadratic function by addition:

For instance, it could be:

$f(x)=5 \\ \\ g(x)=-3x^2 \\ \\ \\ j(x)=f(x)+g(x) \\ \\ j(x)=5-3x^2\\ \\ j(x)=-3x^2+5$

Recall that a constant function like f(x) is also a linear function.

3. Two linear functions by multiplication

Applying a similar method we can conclude that the product of two linear functions will lead to the resulting function j(x). The product can be written as:

$j(x)=f(x)g(x)$

Since both functions have degree 1, then when multiplying them we will get a maximum degree of 2.