Constant Function

Definition

A constant function is a function where the output value is the same for any input value. Regardless of the input, the function always returns the same constant value, making it a special type of linear function and polynomial function.

5 Must Know Facts For Your Next Test

1. The graph of a constant function is a horizontal line, as the $y$-value is the same for all $x$-values.
2. Constant functions have a slope of zero, as the change in $y$-value is always zero for any change in $x$-value.
3. Constant functions are the simplest type of linear function, where the linear coefficient $m$ is zero.
4. Constant functions are a special case of polynomial functions, where the degree of the variable term is zero.
5. Constant functions are often used to model situations where a quantity does not change, such as a fixed cost or a constant interest rate.

Review Questions

• Explain how a constant function is a specific type of linear function.
  • A constant function is a special case of a linear function, where the slope $m$ is zero. In a linear function, the output $y$ is linearly related to the input $x$ through the equation $f(x) = mx + b$. When the slope $m$ is zero, the equation simplifies to $f(x) = b$, meaning the output is a constant value $b$ regardless of the input $x$. This makes a constant function a unique type of linear function with a constant, unchanging output.
• Describe the relationship between constant functions and polynomial functions.
  • Constant functions are a specific type of polynomial function. A polynomial function is a function that can be expressed as the sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power. In a constant function, the variable term has a power of zero, meaning the function can be written as $f(x) = c$, where $c$ is a constant. This makes a constant function the simplest type of polynomial function, where the degree of the variable term is zero and the output is always a fixed value.
• Analyze the key characteristics of a constant function and explain how they differ from other types of functions.
  • The defining characteristic of a constant function is that the output value is the same for any input value. This means the function has a constant, unchanging output, regardless of the input. This is in contrast to other types of functions, such as linear functions and polynomial functions, where the output value changes as the input value changes. Constant functions have a slope of zero, meaning there is no change in the $y$-value for any change in the $x$-value. This makes the graph of a constant function a horizontal line, unlike the linear or curved graphs of other function types. Overall, the constancy of the output value is the key feature that distinguishes constant functions from other more dynamic function types.

Related terms

Linear Function: A linear function is a function that can be expressed in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. A constant function is a specific type of linear function where the slope $m$ is zero.

Polynomial Function: A polynomial function is a function that can be expressed as the sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power. A constant function is a specific type of polynomial function where the variable term has a power of zero.

Equation: An equation is a mathematical statement that asserts the equality of two expressions. The equation for a constant function is $f(x) = c$, where $c$ is the constant value that the function always returns.