Question 2.8.1: Using Operations on Functions Let ƒ(x) = x² + 1 and g(x) = 3...
Using Operations on Functions
Let ƒ(x) = x² + 1 and g(x) = 3x + 5. Find each of the following.
(a) (ƒ + g)(1) (b) (ƒ – g)(-3) (c) (ƒg)(5) (d) ( \frac{ƒ}{g})(0)
(a) First determine ƒ(1) = 2 and g(1) = 8. Then use the definition.
(ƒ + g)(1)
= ƒ(1) + g(1) (ƒ + g)(x) = ƒ(x)+ g(x)
= 2 + 8 ƒ(1) = 1² + 1; g(1) = 3(1) + 5
= 10 Add.
(b) (ƒ – g)(-3)
= ƒ(-3) – g(-3) (ƒ – g)(x) = ƒ(x) – g(x)
= 10 – (-4) ƒ(-3) = (-3)² + 1; g(-3) = 3(-3) + 5
= 14 Subtract.
(c) (ƒg)(5)
= ƒ(5) • g(5) (ƒg)(x) = ƒ(x) • g(x)
= (5²+ 1)(3 • 5 + 5) ƒ(x) = x² + 1; g(x) = 3x + 5
= 26 • 20 ƒ(5) = 26; g(5) = 20
= 520 Multiply.
(d) (\frac{ƒ}{g})(0)
= \frac{ƒ(0) }{g(0)} (\frac{ƒ}{g})(x) = ƒ\frac{(x) }{g(x)}
= \frac{0² + 1}{3(0) + 5} ƒ(x) = x² + 1
g(x) = 3x + 5
= \frac{1}{5} Simplify.