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Solving with
Intersections

Discover why the x-coordinates of intersection points give us the solutions to \( f(x) = g(x) \), and learn to find them graphically, numerically, and algebraically.

Algebra 2

What Does \( f(x) = g(x) \) Mean?

When we write \( f(x) = g(x) \), we are asking: for which input values \( x \) do two different functions produce the same output?

This idea connects algebra (solving equations) to geometry (graphing curves). Every solution to \( f(x) = g(x) \) corresponds to a point where the two graphs cross.

The Graphical Connection

x y
\( x_1 \)
\( x_2 \)
\( f(x) \)
\( g(x) \)

Why the x-Coordinates?

When we solve \( f(x) = g(x) \), we want to find the input values that make the equation true. The y-coordinate tells us the shared output, but the x-coordinate is the solution to the equation.

x-coordinate
The solution to \( f(x) = g(x) \). This is the value we report.
y-coordinate
The shared output: \( f(x) = g(x) = y \). Confirms the match, but is not the solution.

The Core Principle

1
Same x, same y: at an intersection point \((a, b)\), both \( f(a) = b \) and \( g(a) = b \).
2
Therefore \( f(a) = g(a) \): since both equal \( b \), by the transitive property.
3
Conversely: if \( f(a) = g(a) \), then the point \((a, f(a))\) lies on both graphs, creating an intersection.

Visualizing the Equivalence

x y
\( x = -1 \)
\( x = 2 \)
\( y = x^2 \)
\( y = x + 2 \)
 (-1, 1) (2, 4)

Three Methods to Find Solutions

Graphing
Plot both functions and read off the x-coordinates of intersection points. Best with technology (Desmos, graphing calculator).
Tables
Make a table of values for both \( f(x) \) and \( g(x) \). Where the outputs match, you have found a solution.
Algebra
Set \( f(x) = g(x) \) and solve the resulting equation. Gives exact answers when possible.

Exact vs. Approximate Solutions

Exact (Algebraic) Approximate (Graphical/Table)
Method Set \( f(x) = g(x) \), solve for x Graph or table: read intersection
Precision Perfect: \( x = \frac{-1+\sqrt{5}}{2} \) Depends on zoom: \( x \approx 0.618 \)
Best for Linear, polynomial, simple rational Mixed types: \( 2^x = 3x \)
Limitation May be impossible to solve by hand May miss solutions outside viewing window

How to Apply This Method

1
Identify both sides. Rewrite the equation as \( f(x) = g(x) \) by assigning each side its own function.
2
Choose a method. Decide whether to solve algebraically, graph both functions, or build a table.
3
Find intersections. Locate where \( f(x) = g(x) \) by finding points that lie on both graphs.
4
Read x-coordinates. The x-values at each intersection point are the solutions to the original equation.
5
Verify. Substitute each solution back into both \( f(x) \) and \( g(x) \) to confirm equal outputs.

Example 1: Linear Meets Linear

Solve \( 2x + 1 = -x + 7 \) by finding where the two lines intersect.

x y 1 2 3 (2, 5)
\( y = 2x + 1 \)
\( y = -x + 7 \)

Example 2: Quadratic Meets Linear

Solve \( x^2 - 1 = 2x - 1 \) by graphing \( f(x) = x^2 - 1 \) and \( g(x) = 2x - 1 \).

x y
\( x = 0 \)
\( x = 2 \)
 (0, -1) (2, 3)
\( f(x) = x^2 - 1 \)
\( g(x) = 2x - 1 \)

Example 3: Exponential Meets Linear

Solve \( 2^x = x + 3 \). This cannot be solved algebraically, so we use a graph.

x y 1 2 3 -1
\( x \approx 2.47 \)
\( y = 2^x \)
\( y = x + 3 \)

Finding Solutions with a Table

Solve \( |x - 1| = \frac{1}{2}x + 1 \) using a table of values.

\( x \) \( f(x) = |x-1| \) \( g(x) = \frac{1}{2}x + 1 \) Equal?
-230No
-120.5No
011✓ Yes
101.5No
212No
322.5No
433✓ Yes

This Works for All Function Types

Linear
\( 3x + 2 = -x + 6 \)
Polynomial
\( x^3 = 4x \)
Exponential
\( 2^x = x + 5 \)
Logarithmic
\( \ln(x) = x - 2 \)
Absolute Value
\( |x - 3| = 2x \)
Rational
\( \frac{1}{x} = x - 1 \)

Common Mistakes

✕ Reporting the y-coordinate
The solution to \( f(x) = g(x) \) is the x-value, not the y-value of the intersection point.
✕ Missing solutions
A small graphing window may hide intersections. Always consider the full domain and zoom out.
✕ Confusing "no solution" with "hard to solve"
Just because algebra is difficult does not mean solutions do not exist. Graph to check before concluding "no solution."

Summary of Methods

Method How It Works When to Use
Algebraic Set \( f(x) = g(x) \), solve for x Both functions are same type (e.g., polynomial = polynomial)
Graphical Graph both, read intersection x-values Mixed types or complex functions
Table Evaluate both at many x-values, find matches Integer solutions or successive approximation
Technology Use "intersect" feature on calculator or Desmos Need precise decimal approximations

Question 1 of 5

The graphs of \( y = f(x) \) and \( y = g(x) \) intersect at the point \( (3, 7) \). What can you conclude?

Question 2 of 5

Which equation is solved by finding where \( y = x^2 \) and \( y = 3x + 4 \) intersect?

Question 3 of 5

A student graphs \( y = 2^x \) and \( y = 5 \) and sees they intersect at approximately \( (2.32, 5) \). What is the approximate solution to \( 2^x = 5 \)?

Question 4 of 5

The equation \( |x| = x^2 - 2 \) has how many solutions? (Hint: think about the graphs of \( y = |x| \) and \( y = x^2 - 2 \).)

Question 5 of 5

A table shows that \( f(2) = 6 \), \( g(2) = 4 \), \( f(3) = 5 \), and \( g(3) = 5 \). At what value of \( x \) must the equation \( f(x) = g(x) \) have a solution?