Please help with this problem.  Please explain each step. 1.  We define the following functions: f(x) = 2x + 5          g(x) = x^2 - 3       h(x) = 7-x/3     Compute (f - h)(4)...

Please help with this problem.  Please explain each step.

1.  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: (f o g) (x)

    B: (h o g)(x)

Find the inverse functions:

    C:  f^-1(x)            D: h^-1(x)

Asked on by gturner49

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embizze | High School Teacher | (Level 1) Educator Emeritus

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We are given ```f(x)=2x+5,g(x)=x^2-3,h(x)=(7-x)/3`

(1) Find (f-h)(4):

`(f-h)(4)=f(4)-h(4)=2(4)+5-(7-4)/3=13-1=12`

Alternatively, we can compute f-g and then evaluate at x=4:

`f-g=2x+5-(7-x)/3=(6x)/3+15/3-(7-x)/3=(7x+8)/3` ; evaluated at x=4 we get `(7(4)+8)/3=36/3=12` as before.

(2) (a) Find `(f circ g)(x)` : This is equivalent to finding f(g(x)).

** Note that f(x)=2x+5, so f(4)=2(4)+5 and f(-1)=2(-1)+5 and f(w)=2(w)+5 so f(g(x))=2(g(x))+5 **

`f(g(x))=2(g(x))+5=2(x^2-3)+5=2x^2-6+5=2x^2-1`

(b) Find ```(h circ g)(x)=h(g(x))`

`h(g(x))=(7-g(x))/3=(7-(x^2-3))/3=(-x^2+10)/3`

(3) To find the inverse of a function, one method is to write the function as y="stuff"; then exchange x and y, and solve the new function for y. This new function is the inverse of the original.

(a) Find `f^(-1)(x)` (Note that this is the inverse of f(x), not the reciprocal which would be written `(f(x))^(-1)` )

f(x)=2x+5
y=2x+5   Swap x and y
------------

x=2y+5  This is the inverse function -- solve for y to put in function form:
2y=x-5
`y=(x-5)/2=1/2(x-5)=x/2-5/2`

So `f^(-1)(x)=(x-5)/2`

** Consider what happened to x in the function f: first you took the input and multiplied by 2, then you added 5 to the result. The inverse function should "undo" these operations. Much as in the morning you put on socks then shoes, in the evening you undo these operations in the reverse order thus removing shoes then socks.

So the inverse function will take the input, subtract 5 (undoing the add 5) and then divide the result by 2 (thus undoing the multiplication by 2.) **

*** The graphs of the function and the inverse are reflections about the line y=x; here the graph of f is black and the inverse in red:

**** To verify that the functions are inverses, note that `f(f^(-1)(x))=f^(-1)(f(x))=x`

`f(f^(-1)(x))=2((x-5)/2)+5=x`  and `f^(-1)(f(x))=(2x+5-5)/2=x`

(b) Find `h^(-1)(x)` :

`h(x)=(7-x)/3`

`y=(7-x)/3`
------------- Exchange x and y and solve for y:

`x=(7-y)/3`

`7-y=3x`

`y=7-3x`

So `h^(-1)(x)=-3x+7`

Sources:

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