Tutor profile: Blake S.
If $$f(x)=2x^2$$ and $$g(x)=x+2$$, compose these functions and find $$f(g(x))$$ and $$g(f(x))$$.
To find $$f(g(x))$$ we "plug" the function $$g(x)$$ into the function $$f(x)$$ as the variable. When we do this we get the following: $$f(g(x))=2(g(x))^2=2(x+2)^2=2(x^2+4x+4)=2x^2+8x+8$$ To find $$g(f(x))$$ we "plug" the function $$f(x)$$ into the function $$g(x)$$ as the variable. When we do this we get the following: $$g(f(x))=f(x)+2=2x^2+2$$
If the diameter of a cylinder is $$4$$ cm and the height of the cylinder is $$10$$ cm, find the volume of the cylinder.
To find the volume of a cylinder we use the following equation: $$V=$$$$\pi$$$$r^2h$$ Diameter: $$d=2r$$ Radius: $$r=d/2=4/2=2$$ cm Height: $$h=10$$ cm Pi: $$\pi$$$$\approx$$$$3.14$$ To find the volume we then plug these values into the volume equation. $$V=$$$$\pi$$$$r^2h=3.14(2^2)(10)=3.14(4)(10)=125.6$$ $$cm^3$$
Find the discriminant of $$4x^2+3x+1$$ and how many solutions are related.
The discriminant is defined as $$D=b^2-4ac$$ In this example $$a=4$$, $$b=3$$, and $$c=1$$ To find the discriminant, $$D$$, we need to plug these variables into the formula above. After doing this we get $$D=b^2-4ac=3^2-4(4)(1)=9-16=-7$$ The next step is to find how many solutions are related to this equation. The rules for finding how many solutions an equation has are: $$1.$$ if $$D>0$$ then there are $$2$$ Real Solutions. $$2.$$ if $$D=0$$ then there is $$1$$ Real Solution. $$3.$$ if $$D<0$$ then there are $$2$$ Imaginary Solutions. For our equation we found that $$D=-7$$, so using the rules above, we can see that the equation has $$2$$ Imaginary Solutions.
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