Function f and g are defined by
f:x→4-x and g:x→px^2+q where p and q are constants.Given that the composite function gf:x→x^2-8x+11,find
the values of p and q.
f(x) = 4-x
g(x) = px^2 + q
g(f(x))
= g(4-x)
= p(4-x)^2 + q
= p(16 - 8x + x^2) + q
= px^2 - 8p x + 16p+q
if px^2 - 8p x + 16p+q = x^2 - 8x + 11
then p = 1
16p+q = 11
16 + q = 11
q = -5
and -8p = -8
p = 1, which checks out
To find the values of p and q, we need to determine the expressions for f and g and then find their composition gf.
We are given that f(x) = 4 - x and g(x) = px^2 + q.
To find the composition gf(x), we substitute g(x) into f(x), i.e., f(g(x)). This can be expressed as f(g(x)) = 4 - (px^2 + q).
Now we have the expression for gf(x) as gf(x) = x^2 - 8x + 11.
To determine the values of p and q, we equate the coefficients of the corresponding powers of x in gf(x) and f(g(x)).
Comparing the coefficients of x^2, we get: 1 = p.
Comparing the coefficients of x, we get: -8 = 0 (since f(x) = 4 - x and g(x) does not have an x term).
Comparing the constant terms, we get: 11 = 4 - q.
From the second equation, we can see that -8 = 0, which is not possible. Therefore, there are no values of p and q that satisfy the given conditions for the given composite function gf(x) = x^2 - 8x + 11.