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QuestionWhich function has a horizontal asymptote at 4? A. f(x)=2(3)x+4f(x)=2(3)^{x}+4 B. f(x)=2x−4f(x)=2 x-4 C. f(x)=3(2)x−4f(x)=3(2)^{x}-4 D. f(x)=−3x+4f(x)=-3 x+4

Studdy Solution

STEP 1

Assumptions1. A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. . The function is in the form f(x)=a⋅bx+cf(x) = a \cdot b^x + c, f(x)=a⋅x+bf(x) = a \cdot x + b, or f(x)=−a⋅x+bf(x) = -a \cdot x + b where a, b, and c are constants.
3. For the function f(x)=a⋅bx+cf(x) = a \cdot b^x + c, the horizontal asymptote is at y = c.
4. For the function f(x)=a⋅x+bf(x) = a \cdot x + b or f(x)=−a⋅x+bf(x) = -a \cdot x + b, the horizontal asymptote is at y = b if a is0, otherwise there is no horizontal asymptote.

STEP 2

Now, we will check each option to see which function has a horizontal asymptote at4.
Option A f(x)=2()x+4f(x)=2()^{x}+4
Here, the function is in the form f(x)=a⋅bx+cf(x) = a \cdot b^x + c. According to our assumptions, the horizontal asymptote is at y = c. In this case, c =4.

STEP 3

Option B f(x)=2x−f(x)=2x-
Here, the function is in the form f(x)=a⋅x+bf(x) = a \cdot x + b. According to our assumptions, the horizontal asymptote is at y = b if a is0. In this case, a is not0, so there is no horizontal asymptote.

STEP 4

Option C f(x)=3(2)x−4f(x)=3(2)^{x}-4
Here, the function is in the form f(x)=a⋅bx+cf(x) = a \cdot b^x + c. According to our assumptions, the horizontal asymptote is at y = c. In this case, c = -4.

STEP 5

Option D f(x)=−3x+4f(x)=-3x+4
Here, the function is in the form f(x)=−a⋅x+bf(x) = -a \cdot x + b. According to our assumptions, the horizontal asymptote is at y = b if a is0. In this case, a is not0, so there is no horizontal asymptote.

STEP 6

From the above steps, we can see that only Option A f(x)=2(3)x+4f(x)=2(3)^{x}+4 has a horizontal asymptote at4.
So, the correct answer is Option A f(x)=2(3)x+4f(x)=2(3)^{x}+4.

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