The bar graph shows your chances of surviving to various...

The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age $x .$ Use this information to solve Exercises $101-102$ a. Find and interpret $f(90)$ b. Find and interpret $g(90)$ c. Which function serves as a better model for the chance of surviving to age $90 ?$

The bar graph shows your chances of surviving to various ages once you reach 60
The functions
$$
\begin{aligned}
f(x) &=-2.9 x+286 \\
\text { and } g(x) &=0.01 x^{2}-4.9 x+370
\end{aligned}
$$
model the chance, as a percent, that a 60 -year-old will survive to age $x .$ Use this information to solve Exercises $101-102$
a. Find and interpret $f(90)$
b. Find and interpret $g(90)$
c. Which function serves as a better model for the chance of surviving to age $90 ?$

Key Concepts

Comparing Mathematical Models

This concept refers to the process of analyzing different mathematical functions to determine which one best represents a particular set of real-world data. Comparisons are made by evaluating how closely the predictions of each model align with observed outcomes, helping to select the most appropriate and reliable model for a given situation.

Function Evaluation

Evaluating a function involves substituting a specific input value into the function to compute an output. This process allows one to interpret and understand the behavior of the function at particular points, and it is crucial for making predictions about the modeled scenario.

Function Modeling

This concept involves using mathematical functions to represent relationships between variables in real-world scenarios. In many contexts, functions allow us to describe how one quantity changes in relation to another, enabling predictions and inferences about the system being modeled.

Linear Functions

A linear function is one in which the rate of change between the independent variable and the dependent variable is constant, represented by a straight line on a graph. Such functions are often used when the relationship between variables can be approximated by a constant rate of change, making them useful to quickly estimate outcomes over a range of input values.

Quadratic Functions

A quadratic function features a squared term and produces a parabolic graph. These functions are useful for modeling situations where the relationship between variables changes in a non-linear way, often exhibiting a maximum or minimum value. They offer more flexibility than linear models when dealing with phenomena that have accelerating or decelerating trends.

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