Find the Input

Part of the Graphs & Systems section of Coddy's Algebra journey. Lesson 36 of 61.

You can also ask the question the other way round. Which input gives a certain output? For $f\left(x\right) = 3x+2$ , to find $x$ with $f\left(x\right) = 11$ , replace $f\left(x\right)$ by its rule and solve the equation:

$3x+2 = 11$

$3x = 9$

$x = 3$

Evaluating goes from an input to its output. Solving goes from an output back to its input. Both use the same rule, in opposite directions.

The only new step is replacing $f\left(x\right)$ by the rule it stands for. After that, you solve the equation the same way as in the first section.

Challenge

Medium

For $f\left(x\right) = 4x-1$ , find the input with $f\left(x\right) = 19$ .

Try it yourself

f\left(x\right) = 19

Get $x$ alone

Given $f\left(x\right) = 4x-1$

All lessons in Graphs & Systems

1Points and Rules

Two Numbers, One Point A Rule Makes Points The Points Line Up

4Slope-Intercept Form

y = mx + b The Starting Height Write the Equation Solve for y Draw From Any Form Recap: y = mx + b

7Parallel & Perpendicular

Same Slope, Never Meet The Perpendicular Slope A Right-Angle Line Recap: Two Relations

10Systems by Graphing

Two Lines, One Answer Solve by Drawing Check the Crossing When Lines Never Cross Recap: By Graphing

2The Whole Plane

The Four Quadrants Mirror Points On the Line or Not?Recap: Rule to Line

5Point-Slope Form

A Point and a Slope From Two Points Flat or Vertical Through Draw From a Point Recap: Point-Slope

8Function Notation

The Name of a Rule Inputs With SignsFind the InputThe Graph of f Recap: f(x)

11Solving Systems Exactly

Substitution Isolate First Elimination Scale, Then Add Three Unknowns Two Numbers, Two Facts Recap: Two Methods