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# Evaluting Function On A Snow Day

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Use the functions below to answer the questions

g(x)=x2-4x+2                                 f(x)=5x-9                                     h(x)=(1/4)x+6

1. (h-f)(4)
2. Find x if h(x)=9
3. Find n if f(n)=f(3n+1)

Thanks

Mar 21, 2018

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1. $$(h-f)(4)=h(4)-f(4)$$

The left hand side and the right hand of the equation are equivalent. The notation on the right might be more intuitive, though.

 $$h(4)-f(4)$$ Evaluation both functions when x equals 4. $$h(4)=\frac{1}{4}*4+6$$ Here, I have replaced all instances of an "x" with a 4. Now, let's evaluate h(4). $$h(4)=1+6=7$$ Now, let's find f(4). $$f(4)=5*4-9$$ Yet again, every appearance of "x" is replaced with the input, 4. $$f(4)=11$$ The original question wants you to subtract the two functions, so let's do that. $$h(4)-f(4)\\ 7\hspace{6mm}-\hspace{3mm}11$$ $$-4$$

2) If h(x)=9, then we can use substitution to find the value of x:

 $$h(x)=\frac{1}{4}x+6$$ Replace h(x) with 9 since they are equal. $$9=\frac{1}{4}x+6$$ Subtract 6 from both sides of the equation. $$3=\frac{1}{4}x$$ Multiply by 4 on both sides to isolate the variable. $$x=12$$

3) If f(n)=f(3n+1), then we can evaluate both functions for the given input and set them equal to each other.

 $$f(x)=5x-9$$ $$f(x)=5x-9$$ $$f(n)=5n-9$$ $$f(3n+1)=5(2n+1)-9$$ $$f(3n+1)=10n+5-9$$ $$f(n)=5n-9$$ $$f(3n+1)=10n-4$$

As aforementioned, these values are equal, so let's set them equal

 $$\hspace{4mm}f(n)=f(3n+1)\\ 5n-9=10n-4$$ Now, solve for n. Move the constants and linear terms over to one side of the equation. $$-5=5n$$ Finally, divide by 5. $$-1=n$$
Mar 22, 2018