Use the functions below to answer the questions
g(x)=x2-4x+2 f(x)=5x-9 h(x)=(1/4)x+6
Thanks
+2440 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. |
| \(-1=n\) | |
