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2 years ago

If fog(x)=2x-1/x and g(x)=5x+2 , find f(x).

Mathematics

1 answer:

Vikentia [17]2 years ago

8 0

Answer:

Step-by-step explanation:

To do this, we will use a u substitution. Namely, let 5x + 2 = u and solve for x so u is in terms of x, since we have x in our composition.

5x + 2 = u, then

5x = u - 2 and

$x=\frac{u-2}{5}$

which changes f(g(x)) into f(u) {since g(x) = 5x + 2 = u}:

$f(u)=\frac{2(\frac{u-2}{5})-1 }{\frac{u-2}{5} }$ and of course that mess needs to be simplified:

$f(u)=\frac{\frac{2u-4}{5}-1 }{\frac{u-2}{5} }$ and

$f(u)=\frac{\frac{2u-4}{5}-\frac{5}{5} }{\frac{u-2}{5} }$ and

$f(u)=\frac{\frac{2u-4-5}{5} }{\frac{u-2}{5} }$ and

$f(u)=\frac{\frac{2u-9}{5} }{\frac{u-2}{5} }$ and bring up the lower fraction and flip it to multiply to get

$f(u)=\frac{2u-9}{u-2}$ and now we can change the u to an x to get that

$f(x)=\frac{2x-9}{x-2}$

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A circle is growing so that the radius is increasing at the rate of 3 cm/min. How fast is the area of the circle changing at the

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Answer:

The area is growing at a rate of $\frac{dA}{dt} =226.2 \,\frac{cm^2}{min}$

Step-by-step explanation:

<em>Notice that this problem requires the use of implicit differentiation in related rates (some some calculus concepts to be understood), and not all middle school students cover such.</em>

We identify that the info given on the increasing rate of the circle's radius is 3 $\frac{cm}{min}$ and we identify such as the following differential rate:

$\frac{dr}{dt} = 3\,\frac{cm}{min}$

Our unknown is the rate at which the area (A) of the circle is growing under these circumstances,that is, we need to find $\frac{dA}{dt}$ .

So we look into a formula for the area (A) of a circle in terms of its radius (r), so as to have a way of connecting both quantities (A and r):

$A=\pi\,r^2$

We now apply the derivative operator with respect to time ( $\frac{d}{dt}$ ) to this equation, and use chain rule as we find the quadratic form of the radius:

$\frac{d}{dt} [A=\pi\,r^2]\\\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}$

Now we replace the known values of the rate at which the radius is growing ( $\frac{dr}{dt} = 3\,\frac{cm}{min}$ ), and also the value of the radius (r = 12 cm) at which we need to find he specific rate of change for the area :

$\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}\\\frac{dA}{dt} =\pi\,*2*(12\,cm)*(3\,\frac{cm}{min}) \\\frac{dA}{dt} =226.19467 \,\frac{cm^2}{min}\\$