Given the polynomial functions, find the product function and the specified value.
Let f(x) = x + 2 and g(x) = x - 10. Find (fg)(x) and (fg)(-5).
To find the product of two polynomial functions, (fg)(x), you need to multiply them together.
Given f(x) = x + 2 and g(x) = x - 10, let's find (fg)(x):
(fg)(x) = f(x) * g(x)
Substituting the given functions:
(fg)(x) = (x + 2) * (x - 10)
Now, simplify the expression by multiplying the two binomials:
(fg)(x) = x * (x - 10) + 2 * (x - 10)
Apply the distributive property:
(fg)(x) = x^2 - 10x + 2x - 20
Combine like terms:
(fg)(x) = x^2 - 8x - 20
So, the product function (fg)(x) is x^2 - 8x - 20.
To find the value of (fg)(-5), simply substitute -5 for x in the product function:
(fg)(-5) = (-5)^2 - 8(-5) - 20
Simplify:
(fg)(-5) = 25 + 40 - 20
(fg)(-5) = 45
Therefore, the value of (fg)(-5) is 45.