d The domain and range of a function describes the possible x- and y-values of the function.
A
a x=0, x=4
B
b x=6
C
c See solution.
D
dDomain of f(x): All real numbers of x
Range of f(x): All real numbers of y Domain of g(x): All real numbers of x Range g(x): y ≥ -7.5
Practice makes perfect
a From the diagram, we see that at point A and B, the graphs intersect. At these intersections, the functions' x-coordinates are the same. This means if we set f(x) and g(x) equal to each other, the solutions should be these points' x-coordinates.
f(x)&= g(x)
12(x-2)^3+1&= 2x^2-6x-3
From the diagram, we know these coordinates.
We have two solutions.
A( 0,-3) &⇒ x= 0
B( 4,5) &⇒ x= 4
By substituting these x-values into the equation, we can show that they work. This means they make the left-hand side and right-hand side equal.
b In the diagram, we see two points of intersection. However, the function's graph continue beyond what we can see. If they cross again outside of the given viewing area, we have more solutions.
To determine if they do, we have to consider what happens when we go further left or further right in the coordinate plane.
If we go further left, we see that g(x) is pointing upwards while f(x) is pointing downwards. Since g(x) is a parabola and f(x) is a third-degree function, there can be no more turning points for either graph. Therefore, no matter how far to the left we go, the graphs will not intersect again.
If we go further right, we see that both graphs are pointing upward. Additionally, since f(x) is of a higher degree than g(x), the graph of f(x) will grow faster than g(x). This means the functions must intercept at one more point. Zooming out in the coordinate plane, we would eventually see this third solution.
To find the x-coordinate, we will rewrite the original equation so that it equals 0 and then factor it.
Examining the left-hand side, we notice that all of the terms contains at least one x This means we can factor the expression by pulling out this x. We will also factor out 0.5 which gives us nicer numbers inside of the parentheses.
0.5x(x^2-10x+24)=0
To find the third solution, we will use the Quadratic Formula on the trinomial inside of the parentheses.
c Examining the diagram, we notice that f(x) crosses the x-axis at about x=0.8.
This is only an approximation. To determine if we were correct, we should substitute it into f(x) and check if it makes the left-hand side and right-hand side equal.
d The domain and range of a function describes the possible x- and y-values of the function. They can be determined by examining the diagram.
Domain
Looking at both f(x) and g(x), we notice that they go from left to right with no interruptions. This means there domain must be all real numbers of x. In other words, we can substitute any value of x and get an output.
f(x):& All real numbers of x
g(x):& All real numbers of x
Range
Similar to the domain, f(x) does not appear to have any restrictions in the vertical direction. It comes from below and continues upward as we move to the right. Therefore its range is all real numbers of y.
f(x):& All real numbers of y
As for g(x), we notice that the parabola's vertex is a minimum. To find its range, we must find the vertex y-coordinate. We can do this by writing it in graphing form.
Graphing Form: & y=a(x-h)^2+k
Vertex:& (h,k)
To write it in graphing form, we have to complete the square.
Now we can identify the vertex.
Function: & y=(x-1.5)^2+(-1.75)
Vertex:& (1.5,-1.75)
The vertex is at (1.5, -1.75) and since it is a minimum the range must be greater than or equal to -1.75.
g(x): y≥-7.5
