Domain, Range, Intercepts, and End Behavior of Exponential Functions
Grade 9 students in California Reveal Math Algebra 1 learn the domain, range, intercepts, and end behavior of basic exponential functions f(x)=ab^x where a>0. The domain is all real numbers, the range is all positive values (y>0), the y-intercept is (0,a), and the horizontal asymptote is y=0. For exponential growth (b>1), f(x) approaches infinity as x increases and approaches 0 as x decreases. For exponential decay (0<b<1), f(x) approaches 0 as x increases and approaches infinity as x decreases. Students apply these properties to functions like f(x)=3(2)^x and g(x)=5(0.25)^x.
Key Concepts
Property For a basic exponential function $f(x) = ab^x$ where $a 0$: Domain: All real numbers, $( \infty, \infty)$ Range: All positive real numbers, $y 0$ or $(0, \infty)$ $y$ intercept: $(0, a)$ Horizontal Asymptote: The $x$ axis, $y = 0$ End Behavior: Growth ($b 1$): As $x \to \infty$, $f(x) \to \infty$; as $x \to \infty$, $f(x) \to 0$ Decay ($0 < b < 1$): As $x \to \infty$, $f(x) \to 0$; as $x \to \infty$, $f(x) \to \infty$.
Examples Example 1: For the exponential growth function $f(x) = 3(2)^x$, the $y$ intercept is $(0, 3)$ and the horizontal asymptote is $y = 0$. The domain is $( \infty, \infty)$ and the range is $y 0$. As $x \to \infty$, $f(x) \to \infty$, and as $x \to \infty$, $f(x) \to 0$. Example 2: For the exponential decay function $g(x) = 5(0.25)^x$, the $y$ intercept is $(0, 5)$ and the horizontal asymptote is $y = 0$. The domain is $( \infty, \infty)$ and the range is $y 0$. As $x \to \infty$, $g(x) \to 0$, and as $x \to \infty$, $g(x) \to \infty$.
Explanation The domain of a basic exponential function includes all real numbers because you can raise a positive base to any exponent. However, the range is strictly positive because a positive base raised to any power will never result in zero or a negative number. This creates a horizontal asymptote at $y = 0$, which the graph approaches but never touches. The end behavior describes how the graph moves toward infinity on one side and flattens out along the asymptote on the other, depending on whether it represents exponential growth or decay.
Common Questions
What is the domain of a basic exponential function?
The domain of f(x)=ab^x is all real numbers, written as (-infinity, infinity), because you can raise a positive base to any exponent.
What is the range of a basic exponential function?
The range is all positive real numbers, y>0 or (0, infinity), because a positive base raised to any power never results in zero or a negative number.
What is the horizontal asymptote of f(x)=ab^x?
The horizontal asymptote is y=0. The graph approaches but never touches the x-axis because the function values never reach zero.
How do you describe the end behavior of exponential growth?
For growth (b>1): as x approaches infinity, f(x) approaches infinity; as x approaches negative infinity, f(x) approaches 0.
How do you describe the end behavior of exponential decay?
For decay (0<b<1): as x approaches infinity, f(x) approaches 0; as x approaches negative infinity, f(x) approaches infinity.
What is the y-intercept of f(x)=3(2)^x?
The y-intercept is (0, 3) since a=3. This is found by evaluating f(0)=3(2)^0=3(1)=3.