1. Given f(x)=x^2 and g(x)=x+2, determine in standard form:
a) g(f(x))
b) when f(g(x))=g(f(x))
2. If f = {(-10, 1), (-1, -1), (10, 0), (11, 7)} and g = {(-1, -1), (0, 10), (1, -10), (7, 11)} for how many values of x is f x g defined? Explain
3. Given f(x) = -2x + 3 and g(x) = 3x + 4, x ε R, use the sketches of both functions to graph (fg)(x) on the same graph. Show your work.
#1.
g(f) = f+2 = x^2+2
f(g) = g^2 = (x+2)^2
so, when do we have
x^2+2 = (x+2)^2
?
#2.
f and g are both defined only on {-1}
#3.
visit wolframalpha.com
(fg)(x) = -6x^2+x+12
1.
a) To determine g(f(x)), we substitute f(x) into g(x) and simplify.
g(f(x)) = g(x^2) = x^2 + 2
b) To find when f(g(x)) = g(f(x)), we substitute g(x) into f(x) and vice versa, and then equate the two expressions.
f(g(x)) = f(x + 2) = -(x + 2)*2 + 3 = -2x - 4 + 3 = -2x - 1
g(f(x)) = g(x^2) = 3(x^2) + 4 = 3x^2 + 4
We need to find the values of x for which -2x - 1 = 3x^2 + 4.
Rearranging the equation, we get 3x^2 + 2x + 5 = 0.
Since there are no real solutions to this quadratic equation, there are no values of x for which f(g(x)) = g(f(x)).
2. To find how many values of x f x g are defined, we need to determine when there are common x-values between the domains of f and g.
The domain of f consists of the x-values from the given points: -10, -1, 10, and 11.
The domain of g consists of the x-values from the given points: -1, 0, 1, and 7.
To find the common x-values between the two domains, we need to find where the x-values match up in the two sets of points. From the given points, we can see that the common x-values are -1 and 7. Therefore, f x g is defined for two values of x: -1 and 7.
3. To graph (fg)(x) on the same graph, we need to multiply the function values of f(x) and g(x) for each corresponding x-value.
(fg)(x) = f(x) * g(x)
= (-2x + 3) * (3x + 4)
= -6x^2 - 8x + 9x + 12
= -6x^2 + x + 12
Now, we can plot the graph of (fg)(x) by determining the coordinates and connecting the points.
Let's choose several x-values and calculate their corresponding y-values for (fg)(x):
For x = -2: (fg)(-2) = -6(-2)^2 + (-2) + 12 = -24 + (-2) + 12 = -14
For x = -1: (fg)(-1) = -6(-1)^2 + (-1) + 12 = -6 + (-1) + 12 = 5
For x = 0: (fg)(0) = -6(0)^2 + (0) + 12 = 0 + 0 + 12 = 12
For x = 1: (fg)(1) = -6(1)^2 + (1) + 12 = -6 + 1 + 12 = 7
For x = 2: (fg)(2) = -6(2)^2 + (2) + 12 = -24 + 2 + 12 = -10
Plotting these points, we can then draw a curve that represents the graph of (fg)(x) on the same graph as the sketches of f(x) and g(x).