Math  /  Algebra

Question\qquad
4. Find the constant aa that makes f(x)f(x) a continuous functions. A. 0 B. 3 f(x)={x22x+3,x1ax2+4x2,x>1f(x)=\left\{\begin{array}{cc} x^{2}-2 x+3, & x \leq 1 \\ a x^{2}+4 x-2, & x>1 \end{array}\right. C. 2 D. -2

Studdy Solution

STEP 1

1. For a function f(x) f(x) to be continuous at a point x=c x = c , the left-hand limit as x x approaches c c must equal the right-hand limit as x x approaches c c .
2. Additionally, the value of the function at x=c x = c must equal these limits.
3. We need to find the constant a a such that f(x) f(x) is continuous at x=1 x = 1 .

STEP 2

1. Calculate the limit of f(x) f(x) as x x approaches 1 from the left.
2. Calculate the limit of f(x) f(x) as x x approaches 1 from the right.
3. Set these two limits equal to each other to find the value of a a that ensures continuity at x=1 x = 1 .

STEP 3

Calculate the limit of f(x) f(x) as x x approaches 1 from the left.
limx1f(x)=limx1(x22x+3) \lim_{x \to 1^-} f(x) = \lim_{x \to 1} (x^2 - 2x + 3)

STEP 4

Substitute x=1 x = 1 into the expression x22x+3 x^2 - 2x + 3 to find the value of the limit.
limx1f(x)=1221+3=12+3=2 \lim_{x \to 1^-} f(x) = 1^2 - 2 \cdot 1 + 3 = 1 - 2 + 3 = 2

STEP 5

Calculate the limit of f(x) f(x) as x x approaches 1 from the right.
limx1+f(x)=limx1(ax2+4x2) \lim_{x \to 1^+} f(x) = \lim_{x \to 1} (a x^2 + 4x - 2)

STEP 6

Substitute x=1 x = 1 into the expression ax2+4x2 a x^2 + 4x - 2 to find the value of the limit.
limx1+f(x)=a12+412=a+42=a+2 \lim_{x \to 1^+} f(x) = a \cdot 1^2 + 4 \cdot 1 - 2 = a + 4 - 2 = a + 2

STEP 7

Set the left-hand limit equal to the right-hand limit to ensure continuity at x=1 x = 1 .
2=a+2 2 = a + 2

STEP 8

Solve for a a in the equation 2=a+2 2 = a + 2 .
a=22 a = 2 - 2 a=0 a = 0
The value of a a that makes f(x) f(x) a continuous function is 0 \boxed{0} .

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