Unit 2 Mid Chapter
2.4 The ___ is the product of sqrt2 and the length of a side.
Diagonal
2.4 What is the definition of a function?
Each input must have exactly one output.
2.4 Determine whether the equation determines y as a function of x. y^2-x^2=4
y^2-x^2=4 y^2=4+x^2 y=+/-sqrt4+x^2 This is not a function, due to the input having multiple outputs if we plug in any number for x.
2.4 Which of the following is a point on the graph of y=f(x)=x^2-3x+5?
(-2,15). Substitute -2 for x, and 15 for y.
2.4 Find the domain of the function. f(x)=-9x+8
(-infinity,infinity). Since all the values of the variable x result in real numbers as outputs.
2.5 Let g(x)=3x^2-4. Find the average rate of change of the function as x changes from -6 to 2.
-12
2.5 Find the average rate of change of the function f(x)=-3x+6 from a=-1 to b=3
-3
2.5 Determine if the function is even, odd, or neither. f(x)=x/x^11-3x^5
-x/(-x)^11-3(-x)^5 -x/-x11+3x^5 -x/-(x^11-3x^5) x/x^11-3x^5
2.5 Find the average rate of change of the function as x changes from a to b. f(x)=2x^3 ; a=-3, b=1
14
2.5 With the function f(x)=x^2-4x+10, find the simplified expression of... f(4.5+h)-f(4.5)/h
5+h
2.4 Determine whether the equation determines y as a function of x. 6x^2-3y=12
6x^2-3y=12 6x^2-12=3y 2x^2-4=y This equation is a function.
2.5 Extremum points are ___.
A point in which the function is largest or smallest.
2.4 A stone thrown upward with an initial velocity of 160 feet per second will attain a height of h feet in t seconds, described by the equation h(t) shown below. Answer parts (a) through (c). h(t)=160t-16t^2, 0 <= t <= 10 A. What is the domain of h? B.
A. [0,10] B. 336, 384, 144 C. The stone's height when it hits the ground is 0, so replace h(t) with 0 and solve for t. 0 or 10.
2.4 The ___ is the length of a side squared. x^2
Area
2.5 Determine algebraically whether the following function is even, odd, or neither. f(x)=4x^2+5
Even
2.5 Determine algebraically whether the function is even, odd, or neither. f(x)=x/x^7-5x^3
Even.
2.4 True or false? "Every relation is a function"
False. A function is a relation in which no two distinct ordered pairs have the same first component.
2.4 Domain is the set of...
First coordinates
2.4 In the functional notation y=f(x), x is the _______ variable
Independent
2.4 State whether f and g represent the same function. Explain your reason. f(x)=x+5 g(x)=x^2-25/x-5
No, because not only do they not have the same domain, but g is not defined at 5.
2.5 Determine if the function is even, odd, or neither. f(x)=x^9+x^3
Odd
2.4 The ___ is the product of 4 and the length of a side. x*x
Perimeter
2.5 Maximum points are ___.
Point on a curve at which the tangent is either horizontal or vertical.
2.5 Minimum points are ___.
Points at which the value of a function is less than or equal to the value at any nearby point or at any point.
2.5 Turning points are ___.
Points in the graph where the function changes from increasing to decreasing, or decreasing to increasing.
2.4 Range is the set of...
Second coordinates
2.4 Find the domain of the function. f(x)=8/x-9
Set the denominator to 0 to identify all the values of the variable. x-9=0, x=9. If x=9 in this equation, then f(x) is not defined. So, 9 is not part of the domain. Thus, the domain of the function f(x) is (-infinity,9)U(9,inifnity).
2.4 Find the domain of the function. f(x)=x+9/x^2+8x+7
Set the denominator to 0 to identify all the values of the variable. x^2+8x+7=0 Factor the denominator. (x+7)(x+1)=0 Use the zero-product property. x+7=0, x=-7. x+1=0, x=-1. If x equals any of these values, then f(x) is not defined. So, these values are not part of the domain. thus, the domain of the function f(x) is (-infinity,-7)U(-7,-7)U(-1,infinity).
2.4 State the domain and range, and determine if this relation is a function. x y -2 -4 -1 5 0 -3 1 -3 2 3 3 5
The domain is {-2,-1,0,1,2,3}, and the range is {-4,-3,3,5}. This relation is a function.
2.4 State the domain and range, and determine if this relation is a function. w 4 x -> 5 6 y, z -> 7
The domain is {x,y,z}, and the range is {5,7}. This relation is a function.
2.5 True or false? The average rate of change of f as x changes from a to b is the slope of the line through the points (a,(f(a)) and (b,f(b))
True
2.4 Is this relation a function? What is the domain and range? (18,10),(8,20),(24,10)
Yes. The domain is {8,18,24}, and the range is {6,11}
2.4 Let f(x)=2x/sqrt4-x^2. Find each function value. a. f(0) b. f(1) c. f(2) d. f(-2) e. f(-x)
a. 0 b. 2/sqrt3 c. Not defined d. Not defined e. -2x/sqrt4-x^2
2.4 Evaluate the function f(x)=x^2+9x+1 at the given values of the independent variable and simplify. a. f(-9) b. f(x+1) c. f(-x)
a. 1 b. x^2+11x+11 c. x^2-9x+1
2.5 Determine whether each function is even, odd, or neither. f(x)=x^2+5
f(-x)=(-x)^2+5 x^2+5 f(x) Even
2.5 A function f is even if ___.
f(-x)=f(x) for all x in the domain of f
2.4 If the point (9,-14) is on the graph of a function f, then f(9)=___. Why?
f(9)=14. If a point (a,b) is on the graph of f, then a is in the domain of f and then value of "f at a" is b; that is, f(a)=b
2.5 What is the average rate of change?
f(b)-f(a)/b-a
2.5 Let f(x)=2x^2-3x+5 Find and simplify f(x+h)-f(x)/h
f(x)=2x^2-3x+5 Insert f(x+h) =2(x+h)^2-3(x+h)+5 =2(x^2+2xh+h^2)-3x-3h+5 =2x^2+4xh+2h^2-3x-3h+5 Insert this into the original equation. (2x^2+4xh...)-(2x^2-3x+5)/h 2x^2+4xh...-2x^2+3x-5 Combine any like terms, which should leave you with... 4xh+2h^2-3h/h h(4x+2h-3)/h 4x+2h-3 is your final answer.
2.5 Determine whether each function is even, odd, or neither. g(x)=x^3-4x
g(-x)=(-x)^3-4(-x) -x^3+4x Factor out negative -(x^3-4x) =-g(x) Odd
2.5 Determine whether each function is even, odd, or neither. h(x)=2x^3+x^2
h(-x)=2(-x)^3+(-x)^2 -2x^3+x^2 -(2x^3-x^2) Neither
2.4 Determine whether the equation determines y as a function of x. x^11+y^7=2048
x^11+y^7=2048 y^7=2048-x^11 y=^7sqrt2048-x^11 Since the equation is an odd nth root, there is exactly one corresponding y-value in the range for every x-value in the domain. Therefore, y does represent x as a function.