question\nwhich of the following shows the graph of the function below?\n$f(x)=3(3)^x - 2$\nselect the…
question\nwhich of the following shows the graph of the function below?\n$f(x)=3(3)^x - 2$\nselect the correct answer below:
Answer
Explanation:
Step1: Find y-intercept (x=0)
Substitute $x=0$ into $f(x)$: $$f(0)=3(3)^0 - 2 = 3(1) - 2 = 1$$ The graph passes through $(0,1)$.
Step2: Find horizontal asymptote
As $x\to-\infty$, $3^x\to0$, so: $$\lim_{x\to-\infty} f(x) = 3(0) - 2 = -2$$ The horizontal asymptote is $y=-2$.
Step3: Evaluate at x=1
Substitute $x=1$ into $f(x)$: $$f(1)=3(3)^1 - 2 = 9 - 2 = 7$$ The graph passes through $(1,7)$.
Answer:
The first graph (the one with y-intercept at (0,1), approaching y=-2 as x decreases, and passing near (1,7)) is the correct graph.