Problem 93 - Brief Applied Calculus

Question

Can the graph of a function have more than one \(x\) intercept? Can it have more than one \(y\) -intercept?

Step-by-Step Solution

Answer

A function can have multiple \(x\)-intercepts but only one \(y\)-intercept.

Step 1: Identify Definitions

An \(x\)-intercept is where a graph crosses the \(x\)-axis, which means \(f(x) = 0\). A \(y\)-intercept is where the graph crosses the \(y\)-axis, implying the point \((0, f(0))\).

Step 2: Determine Possibility for x-Intercepts

The graph of a function can indeed have more than one \(x\)-intercept. For example, a quadratic function like \(f(x) = x^2 - 1\) has two \(x\)-intercepts, at \(x = 1\) and \(x = -1\).

Step 3: Investigate Possibility for y-Intercepts

A function can only have one \(y\)-intercept because for any given \(x\), there is exactly one corresponding \(y\) in a function (by definition of a function). Thus, the point \( (0, f(0)) \) is unique.

Key Concepts

Functions

Functions are fundamental in calculus and mathematics in general. They describe the relationship between two quantities. A function takes an input, often represented as variable \(x\), and assigns it to exactly one output, \(f(x)\).
Functions can come in many forms, such as linear, quadratic, or exponential.

• A linear function forms a straight line and is expressed as \(f(x) = ax + b\).
• Quadratic functions have the form \(f(x) = ax^2 + bx + c\) and create a parabola.
• Exponential functions grow rapidly and often look like \(f(x) = a^x\).

Understanding functions is key in identifying different points like intercepts on their graphs.

x-intercept

The \(x\)-intercept of a graph is a crucial concept in understanding the behavior of functions. It occurs where the function's graph crosses the \(x\)-axis. This happens when the output value \(f(x)\) equals zero, so we find these intercepts by solving the equation \(f(x) = 0\).
Importantly, a function like \(f(x) = x^2 - 1\) can have more than one \(x\)-intercept. In this case, the graph crosses the \(x\)-axis at two points, \(x = 1\) and \(x = -1\).
This characteristic is especially relevant in polynomial functions, which can have multiple \(x\)-intercepts depending on their degree.

y-intercept

In contrast to the \(x\)-intercept, a function's graph can only have one \(y\)-intercept. This intercept occurs where the graph crosses the \(y\)-axis. To find the \(y\)-intercept, evaluate the function at \(x = 0\), which results in the point \((0, f(0))\).
This result is linked to the definition of a function: one input \(x\) only produces one output \(y\). Thus, the \(y\)-intercept is unique for each function.
Remember that the \(y\)-intercept provides the starting value of \(y\) when \(x = 0\), often serving as an initial or baseline measurement on a graph.

Graph of a Function

Graphs visually represent the behavior of functions, showing how \(y\) values change as \(x\) values vary. Each type of function produces a different shaped graph.

• Linear graphs are straight lines, showing a constant rate of change.
• Quadratic graphs, such as \(f(x) = x^2 - 1\), are parabolas that can open upwards or downwards.
• Exponential graphs show rapid growth or decay.

Interpreting the graph helps us understand crucial points like intercepts and slope. By identifying these attributes, one gains insights into the function's properties and behavior in a much clearer way than solving equations alone.