Question

2. Evaluate the function (3pts each) f(x) = 2x2 -*+5 a) f(-2) b) f(2x) c) f(x-1) 3. Find the domain of the function. (5 pts each) 7X-5 a) f(x) = Vax- x²1 b) g(x) = x25x+6 Domain: Domain: 4. Calculate the difference quotient on the following function. You must show work for credit. (5pts) f(x) = 3x2 - 5​

Answer 1

Answer:

2a : f(-2) = 15

2b : f(2x) = 8x² - 2x + 5

2c : f(x-1) = 2x² - 5x + 8

3a : x > 1/3

3b : x <> {2,3}

4 : 6x

Step-by-step explanation:

2a.

2×(-2)² - (-2) + 5 = 2×4 + 2 + 5 = 15

2b.

2×(2x)² - 2x + 5 = 2×4x² - 2x + 5 = 8x² - 2x + 5

2c.

2×(x-1)² - (x-1) + 5 = 2×(x² - 2x + 1) - x + 1 + 5 =

= 2x² - 4x + 2 - x + 1 + 5 = 2x² - 5x + 8

3a.

the square root is not defined for negative values, which would happen for any x < 1/3. and for x=1/3 the expression would be a division by 0, which is also undefined.

so, all that is left is x>1/3.

3b.

we must avoid a division by 0. everything else is well defined.

so, when is that expression x² - 5x + 6 = 0 ?

solution for a quadratic equation is

x = (-b ± sqrt(b² - 4ac))/2a

a = 1

b = -5

c = 6

x = (5 ± sqrt(25 - 24))/2 = (5 ± sqrt(1))/2 = (5 ± 1)/2

x1 = (5+1)/2 = 6/2 = 3

x2 = (5-1)/2 = 4/2 = 2

so, these values must be avoided for x. everything else is valid.

4.

(f(x+h) - f(x))/h

for h going to 0.

f(x) = 3x² - 5

f(x+h) = 3×(x+h)² - 5

f(x+h) - f(x) = 3×(x² + 2hx + h²) -5 - 3x² + 5 =

= 3x² + 6hx + 3h² - 3x² = 6hx + 3h²

=>

(f(x+h) - f(x))/h = (6hx + 3h²)/h = 6x + 3h = for h 0 = 6x