Answer:
— What is (g × h)(x)?
The answer is 3x²+x-2
— What is g(k(x))?
The answer is 12x - 2 or 2(6x-1)
— What is k(g(0))
The answer is -8
Explanation:
Given these functions —
[tex]f(x) = 5 {x}^{2} - 4x + 1 \\ g(x) = 3x - 2 \\ h(x) = x + 1 \\ k(x) = 4x[/tex]
Find (g × h)(x)
[tex](g \times h)(x) = g(x) \times h(x)[/tex]
Substitute g(x) = 3x - 2 and h(x) = x + 1
[tex](3x - 2) \times (x + 1) \\ (3x - 2)(x + 1)[/tex]
Multiply the polynomial.
[tex]3 {x}^{2} + 3x - 2x - 2[/tex]
Subtract - 2x out of 3x —
[tex]3 {x}^{2} + x - 2[/tex]
Thus, the answer is —
[tex](g \times h)(x) = 3 {x}^{2} + x - 2[/tex]
Find (g(k(x))
Substitute k(x) = 4x in g(x).
[tex]g(x) = 3x - 2 \\ k(x) = 4x[/tex]
[tex]g(k(x)) = g(4x)[/tex]
[tex]g(4x) = 3(4x) - 2[/tex]
Distribute 3 in 4x —
[tex]g(4x) = 12x - 2[/tex]
Thus the answer is —
[tex]g(k(x)) = 12x - 2[/tex]
Alternative Solution
[tex]g(k(x)) = 2(6x - 1)[/tex]
Find k(g(0))
Given two functions — k(x) and g(x)
[tex]k(x) = 4x \\ g(x) = 3x - 2[/tex]
Evaluate the value of g(0) as we substitute x = 0 in g(x)
[tex]g(0 ) = 3(0) - 2 \\ g(0) = 0 - 2 \\ g(0) = - 2[/tex]
Since we need to find k(g(0)), our currently input is g(0).
From k(x) and g(0) —
[tex]k(x) = 4x \\ g(0) = - 2[/tex]
Substitute g(0) = -2 in k(x)
[tex]k(g(0)) = 4(g(0)) \\ k( - 2) = 4( - 2) \\ k( - 2) = - 8[/tex]
Thus, the answer is —
[tex]k(g(0)) = - 8[/tex]
