hello I need help to do this exercise
Consider the function f (x) = (16 x + 33) / (x +3)
1) Determine its domain Df
2) Solve the equation f (x) = 14
3) Show that for all x in Df: f (x) - 15 = (x-12) / (x +3)
4) deduct the resolution of the inequality f (x)> 15.
Thanks for your help.
1 )
The domain of a function is the set of all possible input values , which allows the function formula to work.
The denominator of any fraction cannot have the value zero.
I this case :
x + 3
must be different of zero
x different of - 3
Domain:
( - infinity , - 3 ) U ( - 3 , infinity )
OR
all values of x different of - 3
2 )
( 16 x + 33 ) / ( x + 3 ) = 14 Multiply both sides by ( x + 3 )
16 x + 33 = 14 * ( x + 3 )
16 x + 33 = 14 x + 14 * 3
16 x + 33 = 14 x + 42
16 x - 14 x = 42 - 33
2 x = 9 Divide both sides by 2
x = 9 / 2
3 )
f ( x ) - 15 =
( 16 x + 33 ) / ( x + 3 ) - 15 * ( x + 3 ) / ( x + 3 ) =
( 16 x + 33 - 15 x - 15 * 3 ) / ( x + 3 ) =
( x + 33 - 45 ) / ( x + 3 ) =
( x - 12 ) / ( x + 3 )
4 )
( 16 x + 33 ) / ( x + 3 ) > 15
Multiply both sides by ( x + 3 )
16 x + 33 > 15 * ( x + 3 )
16 x + 33 > 15 x + 15 * 3
16 x + 33 > 15 x + 45
16 x - 15 x > 45 - 33
x > 12
Thanks :)
Of course! I can help you with that.
1) To determine the domain of the function f(x), we need to find all the values of x that make the function defined. The function f(x) has a fraction with a denominator of (x+3). Since dividing by zero is not defined, we need to make sure that x+3 is not equal to zero.
To find the domain, we set the denominator equal to zero and solve for x:
x + 3 = 0
x = -3
So, the function f(x) is defined for all real numbers except x = -3. Therefore, the domain of f(x) is all real numbers except -3.
2) To solve the equation f(x) = 14, we set the function equal to 14 and solve for x:
(16x + 33) / (x + 3) = 14
To solve this equation, we can first cross-multiply to get rid of the fraction:
(16x + 33) = 14(x + 3)
Now we can distribute and solve for x:
16x + 33 = 14x + 42
2x = 9
x = 9/2 or x = 4.5
So the solutions to the equation f(x) = 14 are x = 9/2 or x = 4.5.
3) To show that for all x in Df: f(x) - 15 = (x-12) / (x + 3), we can substitute the function f(x) into the equation and simplify:
f(x) = (16x + 33) / (x + 3)
f(x) - 15 = ((16x + 33) / (x + 3)) - 15
To simplify further, we can find a common denominator and combine the fractions:
f(x) - 15 = ((16x + 33) - 15(x + 3)) / (x + 3)
Expanding and simplifying the numerator:
f(x) - 15 = (16x + 33 - 15x - 45) / (x + 3)
f(x) - 15 = (x -12) / (x + 3)
Therefore, for all x in Df, f(x) - 15 = (x - 12) / (x + 3).
4) To deduce the resolution of the inequality f(x) > 15, we can start by finding the critical points where f(x) equals or crosses the value of 15. We can set the function equal to 15 and solve for x:
(16x + 33) / (x + 3) = 15
To solve this equation, we can cross-multiply and simplify:
16x + 33 = 15(x + 3)
16x + 33 = 15x + 45
x = 12
So, x = 12 is the critical point where f(x) = 15.
To resolve the inequality f(x) > 15, we need to consider two cases:
- When x < -3: Since x is less than -3, the denominator (x+3) is negative, which means the whole fraction is negative. Hence, f(x) will always be less than 15. Therefore, there is no solution in this case.
- When x > -3: Since x is greater than -3, the denominator (x+3) is positive, which means the whole fraction is positive. Hence, f(x) can be greater than 15. Therefore, the solution to the inequality f(x) > 15 is the set of all x values greater than -3.
To summarize, the solution to the inequality f(x) > 15 is x > -3.