which of the following could be the graph of $f(x) = \\left(\\frac{2}{5}\\right)^x$?
which of the following could be the graph of $f(x) = \\left(\\frac{2}{5}\\right)^x$?
Answer
Explanation:
Step1: Identify the form of the function
The function $f(x)=\left(\frac{2}{5}\right)^x$ is an exponential - function of the form $y = a^x$ where $a=\frac{2}{5}$ and $0 < a<1$.
Step2: Recall the properties of exponential functions
For an exponential function $y = a^x$ with $0 < a<1$, the function is a decreasing function. When $x = 0$, $y=a^0 = 1$. As $x\rightarrow+\infty$, $y\rightarrow0$, and as $x\rightarrow-\infty$, $y\rightarrow+\infty$.
Step3: Analyze the graphs
The first graph is an increasing exponential function (since it goes up as $x$ increases), so it is not the graph of $y=\left(\frac{2}{5}\right)^x$. The third and fourth graphs are not exponential - like functions (the third has a negative - valued part for positive $x$ and the fourth has a negative - valued part for negative $x$). The second graph is a decreasing function, passes through the point $(0,1)$, and approaches the $x$ - axis as $x$ goes to positive infinity.