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(5 points each) Suppose that f(x) = (3x+4)/(6-5x) Determine the following (#'s 6 - 10) (6) Domain of f. 3x+4 / 6-5x -> f(x) < 6/5 or f(x) > 6/5 x = 6/5 -> 6-5x=0 -> -5x/-5 = -6/-5 -> 6/5 = x (7) f^-1 y = (3x+4)/(6-5x) x = (3y+4)/(6-5y) -> x(6-5y) = (3y+4)/(6-5y) * (6-5y) x(6-5y) = 3y+4 6x-5xy = 3y+4 6x-5xy-6x = 3y+4-6x -5xy-3y = 3y+4-6x-3y -5xy-3y = 4-6x -y(5x+3) = 4-6x -y(5x+3) / -(5x+3) = 4/-(5x+3) - 6x/-(5x+3) y = -(4-6x)/(3+5x) (8) Range of f (Using #9 will be helpful) y = -(4-6x)/(3+5x) -> 3+5x=0 -> 5x/5 = -3/5 -> x = -3/5 f(x) < -3/5 or f(x) > -3/5 (9) Vertical Asymptotes (if any) x = 6/5 * found from #6 (10) Horizontal Asymptotes (if any) y = -3/5 * found from #8

          (5 points each) Suppose that f(x) = (3x+4)/(6-5x)
Determine the following (#'s 6 - 10)
(6) Domain of f.
3x+4 / 6-5x -> f(x) < 6/5 or f(x) > 6/5
x = 6/5 -> 6-5x=0 -> -5x/-5 = -6/-5 -> 6/5 = x
(7) f^-1
y = (3x+4)/(6-5x)
x = (3y+4)/(6-5y) -> x(6-5y) = (3y+4)/(6-5y) * (6-5y)
x(6-5y) = 3y+4
6x-5xy = 3y+4
6x-5xy-6x = 3y+4-6x
-5xy-3y = 3y+4-6x-3y
-5xy-3y = 4-6x
-y(5x+3) = 4-6x
-y(5x+3) / -(5x+3) = 4/-(5x+3) - 6x/-(5x+3)
y = -(4-6x)/(3+5x)
(8) Range of f (Using #9 will be helpful)
y = -(4-6x)/(3+5x) -> 3+5x=0 -> 5x/5 = -3/5 -> x = -3/5
f(x) < -3/5 or f(x) > -3/5
(9) Vertical Asymptotes (if any)
x = 6/5
* found from #6
(10) Horizontal Asymptotes (if any)
y = -3/5
* found from #8

Show more…

(5 points each) Suppose that f(x) = (3x+4)/(6-5x)
Determine the following (#'s 6 - 10)
(6) Domain of f.
3x+4 / 6-5x -> f(x) < 6/5 or f(x) > 6/5
x = 6/5 -> 6-5x=0 -> -5x/-5 = -6/-5 -> 6/5 = x
(7) f^-1
y = (3x+4)/(6-5x)
x = (3y+4)/(6-5y) -> x(6-5y) = (3y+4)/(6-5y) * (6-5y)
x(6-5y) = 3y+4
6x-5xy = 3y+4
6x-5xy-6x = 3y+4-6x
-5xy-3y = 3y+4-6x-3y
-5xy-3y = 4-6x
-y(5x+3) = 4-6x
-y(5x+3) / -(5x+3) = 4/-(5x+3) - 6x/-(5x+3)
y = -(4-6x)/(3+5x)
(8) Range of f (Using #9 will be helpful)
y = -(4-6x)/(3+5x) -> 3+5x=0 -> 5x/5 = -3/5 -> x = -3/5
f(x) < -3/5 or f(x) > -3/5
(9) Vertical Asymptotes (if any)
x = 6/5
* found from #6
(10) Horizontal Asymptotes (if any)
y = -3/5
* found from #8

Added by Linda R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

09/01/2023

Step 1

To find f(x), we need to integrate f'(x) with respect to x: f(x) = ∫4 dx = 4x + C Now, let's find the domain of f(x). Since there are no restrictions on the values of x, the domain is all real numbers: Domain: (-∞, ∞) Now, let's find the vertical asymptotes Show more…

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Suppose that f(x) = (3x+4)/(6-5x). Determine the following (#'s 6 - 10): (6) Domain of f. (7) f^-1 (8) Range of f (Using #9 will be helpful) (9) Vertical Asymptotes (if any) (10) Horizontal Asymptotes (if any)

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Transcript

00:01 Hello everyone, in this question f of x is given.

00:03 We have to, the first part is we have to find the domain of f.

00:09 So, let us write f of x as y equal to 3x plus 4 divided by 6 minus 5x.

00:16 So, for the domain, the denominator should not be equal that is 6 minus 5x not equal to 0.

00:21 So, this implies minus 5x is not equal to minus 6.

00:25 So, this implies x is equal to not equal to 6 by 5.

00:29 Therefore, the domain will be f of x should be greater than 6 by 5 or f of x should be less than 6 by 5.

00:37 So, this is the required domain.

00:39 Then they asked the inverse of f.

00:42 So, let us consider inverse of f as that is f of x is equal to y is equal to 3x plus 4 divided by 6 minus 5x.

00:53 So, let us write this as x is equal to 3y plus 4 divided by 6 minus 5y.

00:59 So, we will be having x into 6 minus 5y equal to 3y plus 4.

01:07 So, this give you 6x minus 5xy minus 3y equal to 4.

01:14 So, taking minus y common, we will be having 5x plus 3 equal to 4.

01:23 Therefore, minus y into 5x plus 3 is equal to 4 minus 6x.

01:29 So, minus y equal to 4 minus 6x divided by 5x plus 3.

01:35 So, moving minus to this side.

01:37 So, 4 minus 6x minus of 5x plus 3...

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