Question

a and b are real numbers.f(x) = x ^ 2 + ax + bg(x) = x ^ 2 + cx + dThe function y = f[g(x)] - g[f(x)] has no zeros, soA. a = c and b = dB. (a not = c) and b = dC. a = c and (b not = d)D. (a not = c) and (b not = d)Thanks!

Answer 1

Explanations:

Given the following functions where a and b are real numbers:

`\begin{gathered} f(x)=x^2+ax+b \\ g(x)=x^2+cx+d \end{gathered}`

Determine the composite functions f[g(x)] and g[f(x)]

`\begin{gathered} f(g(x))=f(x^2+cx+d) \\ f(x^2+cx+d)=(x^2+cx+d)^2+a(x^2+cx+d)+b \\ f(g(x))=(x^2+cx+d)^2+a(x^2+cx+d)+b \end{gathered}`

Similarly for g(f(x))

`\begin{gathered} g(f(x))=g(x^2+ax+b) \\ g(x^2+ax+b)=(x^2+ax+b)^2+c(x^2+ax+b)+d \\ g(f(x))=(x^2+ax+b)^2+c(x^2+ax+b)+d \end{gathered}`

Take the difference of the functions

`\begin{gathered} y=f(g(x))-g(f(x)) \\ y=(x^2+cx+d)^2+a(x^2+cx+d)+b-\lbrack(x^2+ax+b)^2+c(x^2+ax+b)+d\rbrack \\ \end{gathered}`