Answer:
[tex]f(a+1) -f (a-1) = 8 -4a[/tex]
Step-by-step explanation:
Given
[tex]f(x) = 4x - x^2[/tex]
Required:
[tex]f(a+1) -f (a-1)[/tex]
First, we solve for [tex]f(a+1)[/tex]
This is solved by substituting a + 1 for x in f(x)
[tex]f(x) = 4x - x^2[/tex] becomes
[tex]f(a + 1) = 4(a + 1) - (a+1)^2[/tex]
Open bracket
[tex]f(a + 1) = 4a + 4 - (a+1)(a+1)[/tex]
[tex]f(a + 1) = 4a + 4 - (a^2 + a + a+1)[/tex]
[tex]f(a + 1) = 4a + 4 - (a^2 + 2a+1)[/tex]
Open bracket
[tex]f(a + 1) = 4a + 4 -a^2 - 2a -1[/tex]
Collect like terms
[tex]f(a + 1) = 4 - 1 + 4a - 2a -a^2[/tex]
[tex]f(a + 1) = 3 + 2a -a^2[/tex]
Solving for [tex]f(a-1)[/tex]
This is solved by substituting a - 1 for x in f(x)
[tex]f(x) = 4x - x^2[/tex] becomes
[tex]f(a - 1) = 4(a - 1) - (a-1)^2[/tex]
Open bracket
[tex]f(a - 1) = 4a - 4 - (a-1)(a-1)[/tex]
[tex]f(a - 1) = 4a - 4 - (a^2 - a - a+1)[/tex]
[tex]f(a - 1) = 4a - 4 - (a^2 - 2a+1)[/tex]
Open bracket
[tex]f(a - 1) = 4a - 4 -a^2 + 2a- 1[/tex]
Collect like terms
[tex]f(a - 1) = -4 - 1 + 4a + 2a -a^2[/tex]
[tex]f(a - 1) = -5 + 6a -a^2[/tex]
[tex]f(a+1) -f (a-1)[/tex] becomes
[tex]f(a+1) -f (a-1) = 3 + 2a -a^2 - (-5 + 6a -a^2)[/tex]
Open bracket
[tex]f(a+1) -f (a-1) = 3 + 2a -a^2 +5 - 6a + a^2[/tex]
Collect like terms
[tex]f(a+1) -f (a-1) = 3 +5 + 2a - 6a -a^2 + a^2[/tex]
[tex]f(a+1) -f (a-1) = 8 -4a[/tex]
