Functions and Functional Notation
If , f(x)=x2−9
calculate f(4).
7 because,
Substituting in x=4 gives f(4)=42−9=16−9=7.
Given the following piecewise function, evaluate f(−3).
f(x)=
6x+5 x<−3
−5x+2 x≥−3
17
explanation:
because −3 is on the boundary between the pieces of the domain of f, we have to be careful to choose the correct expression to evaluate. Note that −3≥−3, so we evaluate according to that piece of the function. We substitute x=−3 into −5x+2 and find the value is 17, so that is our answer.
Find the slope of the following line:
y =- 3x + 6
-3
y=mx+b
Suppose we have two supply and demand functions as follows D(q)=40−10q,S(q)=−5+20q, where q is the quantity. What is the price of supply, S, if q=3?
55
How many solutions does the following system have?2x−7y= 11
−6x+21y=−30
none = no solution
Given a function h, defined as h (x)=bx−4, where b is a constant,which of the following statements are correct? Select all that apply?
h(3)=3b−4.
h(b)=2b−4.
h(b)=xb−4.
h(b)=b^2−4.
h(3)=3b−4. and h(b)=b^2−4.
Given the following piecewise function, evaluate f(−3).
f(x)=
−4x+3. x<−3
2x−2 x≥−3
-8
explanation: Because −3 is on the boundary between the pieces of the domain of f, we have to be careful to choose the correct expression to evaluate. Note that −3≥−3, so we evaluate according to that piece of the function. We substitute x=−3 into 2x−2 and find the value is −8, so that is our answer.
Find the slope of the line through the points (−3,7) and (−9,2)
5/6
because m= y2-y1/x2-x1
A heat wave is expected next week. The temperature today is 65∘ F. Next week it is forecasted to be 99∘ F. What is the forecasted high temperature in degrees Celcius?
37.2
solve the following system of equations.
y=−x+8
4x−3y=−3 Give your answer as an ordered pair (a,b)
(3,5)
because Since we already have y solved in terms of x in the first equation, we can substitute y=−x+8 in the second equation. Doing so, we find4x−3(−x+8)=−3 Now, expanding and collecting like terms, we find4x+3x−24=−37x−24=−37x=21So solving this for x, we find x=3. Plugging this back into the equation for y, we findy=−(3)+8=5So the final answer is (3,5).
Consider the function
k(x)=ax+b
, where a and b are constants. If k(2)=3 and k(3)=5, what is the value of b?
-1 , because
Since k(2)=3, we have 2a+b=3,
and since k(3)=5 we have 3a+b=5.
Solving simultaneously gives a=2,b=−1.
Find the domain of f(x)=√2x−3. (Use interval notation)
Find the y-intercept of the following line
y=1/2x+5
Give your answer as an ordered pair (a,b)
(0,5)
Mary sells bottles of lemonade. The demand function for her lemonade is given by D(q)=100−1.5q,and the supply function is S(q)=0.75q ,where q is the quantity of bottles produced. What is price paid by the consumer when q=20?
70
Evaluate the following ordered triples to determine which is the solution of the system: 4x+y-z=12, 5x+y+4z=21, 4x-y+4z=23
(−4,−3,1)
If we try the various ordered triples, we find that (−4,−3,1) works:
(−4)(−4)+(1)(−3)+(−1)(1)=12
(−5)(−4)+(1)(−3)+(4)(1)=21
(−4)(−4)+(−1)(−3)+(4)(1)=23
The list of ordered pairs below represents a function.{(7,11),(−7,−10),(−11,8),(−3,5)}Find the domain of the function.
−11,−7,−3,7
A sales team estimates that the number of new phones they will sell is a function of the price that they set. They estimate that if they set the price at x dollars, they will sell f(x)=4928−7x phones. Therefore, the company's revenue is x⋅(4928−7x). Find the price x which will maximize the company's revenue.
352 because
We are told that the revenue isx⋅(4928−7x)=−7x2+4928x. This is a quadratic function, so to maximize it we should find the x-value of its vertex. Recall that the x-value of the vertex of the parabola ax2+bx+c is given by the formula −b2a. So in this case, we have a=−7 and b=4928, so the x-value of the vertex is−b2a=−(4928)2(−7)=352
Find the slope of the line through the points (2,8) and (9,−10).
-18/7
Mrs. Dill is planning on baking cakes for a summer festival. In order to do this, she needs to spend $15 on new equipment to bake the cakes, plus she estimates that it will cost her 5 cents for every cake that she bakes. If x is the number of cakes that she makes, use the cost function, C(x), to compute the total cost if she decides to make 150 cakes. Give your answer in dollars and cents.
22.5 because
C(x)=15+0.05x
The total cost is C(150)=15+0.05⋅150=$22.50.
Does the ordered triple (−6,1,−2) satisfy the following system of equations?
4x−y+3z=−31
−3x−y−4z=252
x−y−5z=−3
yes
iven the following piecewise function, evaluate f(3).
f(x)=5x+4x<3
7x+2x≥3
23
Find the equation of the line through (1,−9) which is perpendicular to the line y=−x2−2.Give your answer in the form y=mx+b.
y=2x−11