Question

sketch the graph of a function that satisfies the conditions given below. f(0)=1, lim f(x)=1 as x→±∞, lim f(x)=4 as x→0⁺, and lim f(x)= - 2 as x→0⁻. choose the correct graph below.

Explanation:

Step1: Analyze \(f(0) = 1\)

The function value at \(x = 0\) is 1, so the graph should have a point \((0,1)\).

Step2: Analyze \(\lim_{x

ightarrow\pm\infty}f(x)=1\)
As \(x\) approaches positive or negative infinity, the function approaches 1. So the graph should have a horizontal - asymptote \(y = 1\).

Step3: Analyze \(\lim_{x

ightarrow0^{+}}f(x)=4\)
As \(x\) approaches 0 from the right, the function approaches 4.

Step4: Analyze \(\lim_{x

ightarrow0^{-}}f(x)= - 2\)
As \(x\) approaches 0 from the left, the function approaches - 2.

We check each option:

• Option A: Does not satisfy \(\lim_{x

ightarrow0^{+}}f(x)=4\) and \(\lim_{x
ightarrow0^{-}}f(x)= - 2\).

• Option B: Satisfies \(f(0)=1\), \(\lim_{x

ightarrow\pm\infty}f(x)=1\), \(\lim_{x
ightarrow0^{+}}f(x)=4\) and \(\lim_{x
ightarrow0^{-}}f(x)= - 2\).

• Option C: Does not satisfy \(\lim_{x

ightarrow0^{+}}f(x)=4\) and \(\lim_{x
ightarrow0^{-}}f(x)= - 2\).

• Option D: Does not satisfy \(\lim_{x

ightarrow0^{+}}f(x)=4\) and \(\lim_{x
ightarrow0^{-}}f(x)= - 2\).

Answer:

B.