Here are questions from my two "alter egos", Rachelle and Gusella, better known as Patti, even better known as Courtney. I am supposed to find f(g(x)). F(x)=5x^2minus 2x=3 and g(x)=4x^2 plus 7x-5 The answer I got was 5(4x^2+7x-5)^2+-2x+3. Is this correct? Thanks!!!
I am doing the piecewise function x+3 if x is less than -2
x^2 if -2 is less than or equal to x which is less than 1, and finally, -x+2 if x is greater than one. The chart says that
x x+3 (x,y)
-3 0 -3,0
-2 1 -2,1
How come the three is negative? Is there a rhyme or reason to this? Thanks!!
To find f(g(x)), you need to substitute g(x) into f(x). Let's start with the given functions:
f(x) = 5x^2 - 2x + 3
g(x) = 4x^2 + 7x - 5
Substituting g(x) into f(x), we get:
f(g(x)) = f(4x^2 + 7x - 5)
Now, we need to replace every instance of x in f(x) with the expression g(x) and simplify:
f(g(x)) = 5(4x^2 + 7x - 5)^2 - 2(4x^2 + 7x - 5) + 3
Simplifying further, we first expand the squared term:
f(g(x)) = 5(16x^4 + 56x^3 - 40x^2 + 49x^2 - 70x + 25) - 8x^2 - 14x + 10 + 3
Combining like terms:
f(g(x)) = 80x^4 + 280x^3 - 200x^2 + 245x^2 - 350x + 125 - 8x^2 - 14x + 10 + 3
f(g(x)) = 80x^4 + 280x^3 + (245 - 200 - 8)x^2 + (-350 - 14)x + (125 + 10 + 3)
Simplifying further, we get:
f(g(x)) = 80x^4 + 280x^3 + 37x^2 - 364x + 138
So, the expression f(g(x)) is given by 80x^4 + 280x^3 + 37x^2 - 364x + 138.
Regarding the piecewise function:
The value of f(x) is defined differently for different ranges of x. In this case, if x is less than -2, the expression for f(x) is x + 3. This means that when x is less than -2, the value of f(x) will be obtained by adding 3 to the value of x.
So, when x = -3, as mentioned in the chart, the expression f(x) becomes (-3) + 3 = 0.
Similarly, for other values mentioned in the chart, you need to substitute the given value of x into the respective expression for f(x), according to the range in which it lies.