Can anyone help me with this problem? I need to show all steps and explain what is happening. A graph would be great for a visual too if possible. Thank you for any help.
We define the following functions:
f(x) = 2x + 5, g(x) = x^2 – 3, h(x) = 7 – x / 3 (that is 7-x divided by 3)
o Compute (f – h)(4).
o Evaluate the following two compositions:
A: (fog)(x)
B: (hog)(x)
o Graph the g(x) function and transform it so that the graph is moved 6 units to the right and 7 units down.
o Find the inverse functions:
C: f^-1(x)
D: h^-1(x)
What was unclear about
http://www.jiskha.com/display.cgi?id=1386290597
Other than the (7-x)/3 vs 7 - x/3 difference?
Just make the fix and show us whatcha got.
Certainly! Let's break down each part of the problem step by step.
First, let's compute the expression (f - h)(4). To do this, we substitute the value of 4 into both f(x) and h(x).
f(x) = 2x + 5
h(x) = 7 - x/3
Substituting 4 into f(x), we get:
f(4) = 2(4) + 5
= 8 + 5
= 13
Substituting 4 into h(x), we get:
h(4) = 7 - 4/3
= 7 - 1.33
= 5.67 (rounded to two decimal places)
Now, for (f - h)(4), we subtract the values obtained:
(f - h)(4) = f(4) - h(4)
= 13 - 5.67
≈ 7.33 (rounded to two decimal places)
Next, let's evaluate the two compositions (fog)(x) and (hog)(x).
To find (fog)(x), we substitute g(x) into f(x):
(fog)(x) = f(g(x))
= f(x^2 - 3)
And for (hog)(x), we substitute g(x) into h(x):
(hog)(x) = h(g(x))
= h(x^2 - 3)
To graph the function g(x) and transform it, we move it 6 units to the right and 7 units down. The transformation can be achieved by adding 6 to x and subtracting 7 from the function itself. So the transformed function would be:
g(x) + 7 = (x + 6)^2 - 3 + 7
= (x + 6)^2 + 4
Lastly, let's find the inverse functions of f(x) and h(x).
To find the inverse of a function, we swap the x and y variables and solve for y.
For f(x) = 2x + 5:
1. Replace f(x) with y: y = 2x + 5.
2. Swap x and y: x = 2y + 5.
3. Solve for y: x - 5 = 2y.
(x - 5)/2 = y.
So, the inverse function of f(x) is:
f^-1(x) = (x - 5)/2.
For h(x) = 7 - x/3:
1. Replace h(x) with y: y = 7 - x/3.
2. Swap x and y: x = 7 - y/3.
3. Solve for y: 3x = 7 - y.
y = 7 - 3x.
Therefore, the inverse function of h(x) is:
h^-1(x) = 7 - 3x.
I hope this explanation helps! If you have any further questions, feel free to ask.