# If f(x):=x+4 and g(x):= 4x-1, find a function g such that g x f=h

hala718 | Certified Educator

calendarEducator since 2008

starTop subjects are Math, Science, and Social Sciences

f(x) = (x+4)

g(x) = (4x-1)

h(x) = g(x) x f(x)

= (x+4)*(4x-1)

Open brackets:

==> h(x) = 4x^2 - x + 16x - 4

= 4x^2 + 15x - 4

Then ,

h(x) = 4x^2 + 15x - 4

check Approved by eNotes Editorial

neela | Student

f(x) = x+4

g(x) = x-1

To find a function such that g *  f  = h.

Solution:

The operation

We interpret that The  function h(x) is to be determined and not g which is already known (or given).

We interpret  the operation (or composition)   * as  (i)multiplication (ii) any operation or composition.

(i) * is mutiplying operation.

If a*b = ab  , or a multiplied by b, then h(x) = g*f = (x+4)(x-1) = x^2+3x-4. So h(x) = x^2+3x-4.

(ii) * is any operation or composition of two functions:

If the composition is a*b = a^2-b^2 , then h(x) = [g(x)]^2 - {f(x)]^2 = (x+4)^2-(x-1)^2 = x^2+8x+16 - (x^2-2x+1) = 10x+15.

If the composition is a*b = a - b , then h(x) = g(x) - f(x)  = x+4 -(x-1) = 5.