Answer :
Answer:
[tex](g + f)(2) = \sqrt 3 - 3[/tex]
[tex](\frac{f}{g})(-1) = 0[/tex]
[tex](g + f)(-1) = \sqrt{15}[/tex]
[tex](g * f)(2) = -3\sqrt 3[/tex]
Step-by-step explanation:
Given
[tex]f(x) =1 - x^2[/tex]
[tex]g(x) = \sqrt{11 - 4x[/tex]
See attachment
Solving (a): (g + f)(2)
This is calculated as:
[tex](g + f)(2) = g(2) + f(2)[/tex]
Calculate g(2) and f(2)
[tex]g(2) \to \sqrt{11 - 4 * 2} = \sqrt{3}[/tex]
[tex]f(2) = 1 - 2^2 = -3[/tex]
So:
[tex](g + f)(2) = g(2) + f(2)[/tex]
[tex](g + f)(2) = \sqrt 3 - 3[/tex]
Solving (b): [tex](\frac{f}{g})(-1)[/tex]
This is calculated as:
[tex](\frac{f}{g})(-1) = \frac{f(-1)}{g(-1)}[/tex]
Calculate f(-1) and g(-1)
[tex]f(-1) = 1 - (-1)^2 = 0[/tex]
So:
[tex](\frac{f}{g})(-1) = \frac{f(-1)}{g(-1)}[/tex]
[tex](\frac{f}{g})(-1) = \frac{0}{g(-1)}[/tex]
[tex](\frac{f}{g})(-1) = 0[/tex]
Solving (c): (g - f)(-1)
This is calculated as:
[tex](g + f)(-1) = g(-1) - f(-1)[/tex]
Calculate g(-1) and f(-1)
[tex]g(-1) = \sqrt{11 - 4 * -1} = \sqrt{15}[/tex]
[tex]f(-1) = 1 - (-1)^2 = 0[/tex]
So:
[tex](g + f)(-1) = g(-1) - f(-1)[/tex]
[tex](g + f)(-1) = \sqrt{15} - 0[/tex]
[tex](g + f)(-1) = \sqrt{15}[/tex]
Solving (d): (g * f)(2)
This is calculated as:
[tex](g * f)(2) = g(2) * f(2)[/tex]
Calculate g(2) and f(2)
[tex]g(2) \to \sqrt{11 - 4 * 2} = \sqrt{3}[/tex]
[tex]f(2) = 1 - 2^2 = -3[/tex]
So:
[tex](g * f)(2) = g(2) * f(2)[/tex]
[tex](g * f)(2) = \sqrt 3 * -3[/tex]
[tex](g * f)(2) = -3\sqrt 3[/tex]
