Let fx 2x2 3x and gx x 2 Evaluate the following Expand e

Let f(x) = 2x^2 + 3x and g(x) = x + 2. Evaluate the following. Expand each expression and simple results f(5) f(-5) f(-a) f(a + b) f(g(x)) g(f(x)) Find the domain and range for the following functions: f(x) = |x| f(x) = x^2 + 4 domain: domain range: range f(x) = squareroot x - 5 f(x) = -1/x^2 domain: domain: range: range:

Solution

The image is very faint so the answers are on best effort basis.

2. (a)   f(5) = 2 (5 )² + 3 (5)= 2*25 + 15 = 50 + 15 = 65

    (b) f( -5) = 2( -5²) + 3(-5) = 2*25 – 15 = 50 – 15

    (c)   f(-a) = 2(-a)² + 3(-a) = 2a² -3a

    (d)   f(a + b) = 2(a +b)² + 3(a+b) = 2(a² + 2ab + b²)= 2a²+ 4ab + 2b² + 3a + 3b = 2a²+ 3a + 3b +4ab + 2b²  

     (e)   f(g(x)) = f ( x + 2) = 2 (x +2)² + 3(x+2)= 2(x² + 4x +4) = 3x + 6 = 2x²+8x + 3x + 8 + 6 = 2x²+11x +14

     (f)   g(f(x)) = g(2x² + 3x) = 2x² + 3x +2

3. (a) The domain of the function f(x) = |x| is R, the set of all real numbers or, in interval notation, (-,) as x can take any real value. The range is { y R: y 4} or, in interval notation,[ 4,).

   (b) The domain of the function f(x) = x² +4 is R, the set of all real numbers or, in interval notation, (-,) as x can take any real value.The range is { y R: y 0} or, in interval notation,[0,).

   (c )The domain of the function f(x)= (x-5) is {x R: x 5}or,in interval notation,[5,).The range is { y R: y 0} or, in interval notation,[ 0,).

(d) The domain of the function f(x)= -1/x² is {x R: x 0} or,in interval notation,(-, 0)U(0, ). The range is { y R: y < 0} or,in interval notation, (-, 0)