Answer:
Domain: Set of All Real Numbers. (-∞, ∞)
Range: (0, ∞)
Step-by-step explanation:
We need to evaluate (gof)(x). (gof)(x) means the composition of function g(x) and f(x). In order to find (gof)(x) replace every occurrence of x in g(x) with the expression of f(x) as shown below:
We have to find the domain and range of this composite function.
Domain:
The expression is defined for all Real Numbers. It never gets undefined for any value of x. Hence the Domain is set of All Real Numbers.
Range:
The range of the parent exponential function is from 0 to positive infinity. The expression can be related to as a vertical shifted function by 6 units upward. As a result the range will also be shifted up. So the range of the composite function will be from 6 to positive infinity. In interval notation this would be expressed as: (0, ∞)
Answer: 9
Explantion:
Divide: 6 : 2/
3
= 6/
1
· 3/
2
= 6 · 3/
1 · 2
= 18/
2
= 2 · 9/
2 · 1
= 9
Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 2/
3
is 3/
2
) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the next intermediate step, , cancel by a common factor of 2 gives 9/
1
.
In words - six divided by two thirds = nine.
Option D:
Difference between g(x) and f(x) is 44.
Solution:
Let us first write the function of f(x).
Take any two points on the line (0, -1) and (-2, 0).
Here,
Equation of a line:
Do cross multiplication.
-2(y + 1) = x
-2y - 2 = x
Add 2 on both sides.
-2y = x + 2
Divide by -2 on both sides.
y = -0.5x - 1
f(x) = -0.5x - 1
At x = 20,
f(20) = -0.5(20) - 1 = -11
g(x) = 1.5x + 3
g(20) = 1.5(20) + 3 = 33
Difference between g(x) and f(x)
= g(20) - f(20)
= 33 - (-11)
= 33 + 11
= 44
Difference between g(x) and f(x) is 44.
Option D is the correct answer.