Question

If function f [ x + 1 ] = 3x² - x + 1 ,then f [ x ] = ?

will be greatly appreciated, thanks :)))​

Answer 1

f(x)=3x^2 - x

Step-by-step explanation:

f((x)+1)= (3x^2 - x) + 1

Answer 2

Answer: f(x) = 3x^2-7x+5

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Step-by-step explanation:

Assume that f(x) takes the following form

f(x) = ax^2+bx+c

Then replace each x with x+1

f(x+1) = a(x+1)^2 + b(x+1) + c

Expand out the right hand side and move terms a bit to get this:

f(x+1) = a(x+1)^2 + b(x+1) + c

f(x+1) = a(x^2+2x+1) + b(x+1) + c

f(x+1) = ax^2+2ax+a + bx+b + c

f(x+1) = (ax^2) + (2ax+bx) + (a+b+c)

f(x+1) = ax^2 + (2a+b)x + (a+b+c)

Now equate this with the expression given 3x^2-x+1

ax^2 + (2a+b)x + (a+b+c) = 3x^2 - x + 1

We see that

• a = 3
• 2a+b = -1
• a+b+c = 1

after equating the corresponding coefficients.

Use the value of 'a' to find b in the second equation.

2a+b = -1

2(3)+b = -1

6+b = -1

b = -1-6

b = -7

Then we can find c.

a+b+c = 1

3+(-7)+c = 1

-4+c = 1

c = 1+4

c = 5

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Summary:

We found that

• a = 3
• b = -7
• c = 5

This means we go from f(x) = ax^2+bx+c to f(x) = 3x^2-7x+5

To check this, simplifying f(x+1) = 3(x+1)^2-7(x+1)+5 should get you to f(x+1) = 3x^2-x+1

I'll let you do this part.