Interpreting Function Values in Context

Introduction

Key Concepts

Understanding Functions

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. It is often denoted by $f(x)$, where $x$ represents the input value, and $f(x)$ represents the output value or function value.

Function Notation

Function notation provides a concise way to represent functions. The expression $f(x)$ denotes the function $f$ evaluated at the input $x$. For example, if $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Evaluating Functions

Evaluating a function involves substituting the input value into the function's expression to find the corresponding output value. Consider the function $f(x) = x^2 - 5x + 6$. To find $f(3)$, substitute $x$ with 3: $$ f(3) = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0 $$ Thus, the function value at $x=3$ is 0.

Interpreting Function Values in Context

Interpreting function values requires understanding the real-world meaning behind the input and output variables. For instance, if a function models the cost of producing $x$ items, then $f(x)$ represents the total cost associated with producing $x$ items.

Example: Suppose the total cost $C$ for producing $x$ units of a product is given by the function $C(x) = 50x + 200$. Here, $C(x)$ represents the total cost, $x$ is the number of units produced, $50x$ is the variable cost per unit, and $200$ is the fixed cost. To find the cost of producing 10 units: $$ C(10) = 50(10) + 200 = 500 + 200 = 700 $$ Therefore, producing 10 units costs \$700.

Domain and Range

The domain of a function is the set of all possible input values ($x$) for which the function is defined. The range is the set of all possible output values ($f(x)$). Understanding the domain and range is essential for interpreting function values in specific contexts.

For example, in the cost function $C(x) = 50x + 200$, the domain is $x \geq 0$ because negative production quantities do not make sense in this context. The range would be all values $C(x) \geq 200$, indicating that the total cost cannot be less than the fixed cost.

Real-World Applications

Functions are used to model various real-life situations, such as financial planning, physics problems, biology growth models, and more. Being able to interpret function values in these contexts allows for meaningful analysis and informed decision-making.

Example: Consider the function modeling the height $h(t)$ of a ball thrown upwards from a height of 2 meters with an initial velocity of 20 m/s: $$ h(t) = -5t^2 + 20t + 2 $$ To find the height of the ball at $t = 2$ seconds: $$ h(2) = -5(2)^2 + 20(2) + 2 = -20 + 40 + 2 = 22 \text{ meters} $$ Therefore, the ball is 22 meters high at 2 seconds after being thrown.

Inverse Functions

An inverse function reverses the effect of a given function. If $f(x) = y$, then the inverse function $f^{-1}(y) = x$. Interpreting function values in the context of inverse functions can provide insights into the relationships between variables.

Example: If $f(x) = 3x + 4$, then to find the inverse function, solve for $x$: $$ y = 3x + 4 \\ y - 4 = 3x \\ x = \frac{y - 4}{3} \\ f^{-1}(y) = \frac{y - 4}{3} $$ Thus, the inverse function $f^{-1}(y)$ allows us to determine the original input $x$ given an output $y$.

Composite Functions

A composite function is formed when one function is applied to the result of another. For functions $f(x)$ and $g(x)$, the composite function $f(g(x))$ represents the application of $f$ to the result of $g(x)$.

Example: Let $f(x) = x + 2$ and $g(x) = 3x$. The composite function $f(g(x))$ is: $$ f(g(x)) = f(3x) = 3x + 2 $$ To interpret $f(g(x))$ at $x = 4$: $$ f(g(4)) = f(12) = 14 $$ Therefore, $f(g(4)) = 14$.

Comparison Table

Summary and Key Takeaways