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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
 #1
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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

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