Question

1. describe and correct the error a student made in finding the rule for the composition ( f circ g ) when ( f(x) = 2x^2 - 3x + 1 ) and ( g(x) = 2x - 1 ).

( (f circ g)(x) = f(g(x)) )
( = 2(2x - 1)^2 - 3x + 1 )
( = 2(4x^2 - 4x + 1) - 3x + 1 )
( = 8x^2 - 11x + 3 )

Explanation:

Step1: Identify the student's error

When substituting $g(x)=2x-1$ into $f(x)=2x^2-3x+1$, the student only replaced $x$ with $2x-1$ in the first term, but left the $-3x+1$ unchanged. The correct substitution requires replacing all instances of $x$ in $f(x)$ with $g(x)$.

Step2: Correct substitution into $f(x)$

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

Step3: Expand $(2x-1)^2$

$$(2x-1)^2 = 4x^2 - 4x + 1$$
Substitute back:
$$f(g(x)) = 2(4x^2 - 4x + 1) - 3(2x-1) + 1$$

Step4: Distribute the coefficients

$$= 8x^2 - 8x + 2 - 6x + 3 + 1$$

Step5: Combine like terms

$$= 8x^2 + (-8x -6x) + (2 + 3 + 1)$$
$$= 8x^2 -14x + 6$$

Answer:

The student's error was failing to substitute $g(x)=2x-1$ for all $x$-terms in $f(x)$ (they only substituted the first $x$, leaving $-3x+1$ unchanged). The correct composition is $\boldsymbol{8x^2 -14x + 6}$.