I need help with this question:
Express below function in terms of f(x) if f(x)= x^2
y= 3(-4x-2)^2 + 5
let f(x) = z = x^2
so x = sqrt z and z always positive
y = 3 (-4 sqrt z-2)^2 + 5
y = 3(16 z + 16 sqrt z +4) + 5
y = 48 z + 48 z^.5+ 17
y = 48 f(x) +48 f(x)^.5 + 17
It's kind of confusing. Can you show me without putting z.
believe me, it is much less confusing using another symbol for x^2 then replacing it at the end.
You have
y= 3(-4x-2)^2 + 5
= 3(16x^2+16x+4)+5
= 48x^2+48x+17
Now, f = x^2, so x = √f
That makes
y = 48f + 48√f + 17
To express the function y in terms of f(x), you need to substitute f(x) = x^2 in place of x in the given function.
Given:
y = 3(-4x-2)^2 + 5
Now, replace each occurrence of x with f(x) = x^2:
y = 3(-4f(x)-2)^2 + 5
Next, simplify the expression inside the parentheses:
y = 3(-4f(x)-2)(-4f(x)-2) + 5
Now, distribute and simplify:
y = 3(16f(x)^2 + 16f(x) + 4) + 5
Finally, simplify further:
y = 48f(x)^2 + 48f(x) + 12 + 5
Simplifying the constants:
y = 48f(x)^2 + 48f(x) + 17
Therefore, the function y, expressed in terms of f(x), is:
y = 48f(x)^2 + 48f(x) + 17