When comparing linear, quadratic, cubic, and exponential functions, what key feature do all four function types share?
Understand the Problem
The question is asking to identify a common characteristic shared by linear, quadratic, cubic, and exponential functions. We must evaluate the options given and determine which feature is present across all four function types.
Answer
The domain of all four function types is all real numbers.
Answer for screen readers
The domain of all four function types is all real numbers.
Steps to Solve
- Analyze the domain of each function type
Linear functions, such as $f(x) = x$, have a domain of all real numbers. Quadratic functions, such as $f(x) = x^2$, have a domain of all real numbers. Cubic functions, such as $f(x) = x^3$, have a domain of all real numbers. Exponential functions, such as $f(x) = 2^x$, have a domain of all real numbers. Thus, all four function types have a domain of all real numbers.
- Analyze the range of each function type
Linear functions, such as $f(x) = x$, have a range of all real numbers. Quadratic functions, such as $f(x) = x^2$, have a range of $y \geq 0$, not all real numbers. Cubic functions, such as $f(x) = x^3$, have a range of all real numbers. Exponential functions, such as $f(x) = 2^x$, have a range of $y > 0$, not all real numbers. Thus, not all four function types have a range of all real numbers.
- Analyze the minimum value of each function type
Linear functions, such as $f(x) = x$, do not have a minimum value. Quadratic functions, such as $f(x) = x^2$, can have a minimum value of (0,0). However, $f(x) = x^2 + 1$ also a quadratic function, has minimum value of (0,1). Cubic functions, such as $f(x) = x^3$, do not have a minimum value. Exponential functions, such as $f(x) = 2^x$, do not have a minimum value of (0, 0). Thus, not all four function types have a minimum value of (0, 0).
- Analyze the x-intercept of each function type
Linear functions, such as $f(x) = x$, can have an x-intercept of (0, 0). Not all linear functions have an x-intercept of (0,0), for example, $f(x) = x + 1$ has an x-intercept of (-1, 0). Quadratic functions, such as $f(x) = x^2 + 1$, do not have an x-intercept of (0, 0). Cubic functions, such as $f(x) = x^3 +2$, do not have an x-intercept of (0, 0). Exponential functions, such as $f(x) = 2^x$, never intersect the x-axis, so do not have an x-intercept. Thus, not all four function types have an x-intercept of (0, 0).
The domain of all four function types is all real numbers.
More Information
The domain of a function is the set of all possible input values (x-values) for which the function is defined. Linear, quadratic, cubic, and exponential functions are all defined for any real number x.
Tips
A common mistake is to assume all quadratic functions have a range of all real numbers or to think that exponential functions can take on negative values, which they cannot. For instance, exponential functions of the form $f(x) = a^x$ where $a > 0$, have a range of $(0, \infty)$.
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