Can anyone help me with the following, by providing the steps to get there and what it is that you are doing? any help would be greatly appreciated...
We define the following functions:
f(x)=2x+5 g(x)=x^2-3 h(x)=7-x/3
◦Compute (f – h)(4).
◦Evaluate the following two compositions:
A: (fog)(x)
B: (hog)(x)
◦Graph the g(x) function and transform it so that the graph is moved 6 units to the right and 7 units down.
How do I figure out what these values are for the g(x)?
◦Find the inverse functions:
C: f^-1(x)
D: h^-1(x)
f(x)=2x+5 g(x)=x^2-3 h(x)=7-x/3
(f-h)(4) = f(4)-h(4) = 9-17/3 = 10/3
(f◦g)(x) = f(g) = 2g+5 = 2(x^2-3)+5 = 2x^2-1
(h◦g) = 7-g/3 = 7-(x^2-3)/3 = 8-x^2/3
g(x-6)-7 = (x-6)^2-3-7 = (x-6)^2-10
graph at wolframalpha.com
f-1(x) = (x-5)/2
h-1(x) = 21-3x
To compute (f - h)(4), we need to substitute the value of 4 into the functions f(x) and h(x), and then subtract the results.
Step 1: Substitute 4 into f(x)
f(x) = 2x + 5
f(4) = 2(4) + 5
f(4) = 8 + 5
f(4) = 13
Step 2: Substitute 4 into h(x)
h(x) = 7 - x/3
h(4) = 7 - 4/3
h(4) = 7 - 4/3
h(4) = 21/3 - 4/3
h(4) = 17/3
Step 3: Compute the subtraction
(f - h)(4) = f(4) - h(4)
(f - h)(4) = 13 - 17/3
To evaluate the compositions (fog)(x) and (hog)(x), we need to substitute the functions g(x) and f(x) into each other.
A: (fog)(x)
Step 1: Substitute g(x) into f(x)
f(x) = 2x + 5
g(x) = x^2 - 3
(fog)(x) = f(g(x))
(fog)(x) = f(x^2 - 3)
(fog)(x) = 2(x^2 - 3) + 5
(fog)(x) = 2x^2 - 6 + 5
(fog)(x) = 2x^2 - 1
B: (hog)(x)
Step 1: Substitute g(x) into h(x)
h(x) = 7 - x/3
g(x) = x^2 - 3
(hog)(x) = h(g(x))
(hog)(x) = h(x^2 - 3)
(hog)(x) = 7 - (x^2 - 3)/3
(hog)(x) = 7 - (x^2/3 - 1)
(hog)(x) = 7 - (x^2/3) + 1
(hog)(x) = -x^2/3 + 8
To graph the transformed g(x), we need to move the graph 6 units to the right and 7 units down.
Step 1: Graph the original g(x) function
The original function g(x) = x^2 - 3 can be graphed as a quadratic equation.
Step 2: Transform the graph
To move the graph 6 units to the right, we replace x with (x - 6).
To move the graph 7 units down, we subtract 7 from the entire equation.
So, the transformed function is g(x) = (x - 6)^2 - 3 - 7.
To find the inverse functions, we need to switch the roles of x and y and solve for y.
C: f^-1(x)
Step 1: Switch x and y
x = 2y + 5
Step 2: Solve for y
2y = x - 5
y = (x - 5)/2
So, the inverse function is f^-1(x) = (x - 5)/2.
D: h^-1(x)
Step 1: Switch x and y
x = 7 - y/3
Step 2: Solve for y
y/3 = 7 - x
y = 3(7 - x)
y = 21 - 3x
So, the inverse function is h^-1(x) = 21 - 3x.