Evaluating and Solving Functions

Learning Outcomes

  • Evaluate and solve functions.

In this course, we’ve been using fundamental characteristics of functions to evaluate, solve, identify, transform,and write the equations of each of the toolkit functions. In this module, you’ll study two new functions: the exponential function and the logarithmic function. Their graphs and equations may look a little different from the familiar linear, power, and polynomial functions you’ve seen so far. But they possess the same basic characteristics. That is, they are one-to-one functions, having well-defined corresponding input and output in ordered pairs that represent points in the plane. They are smooth and continuous. And they may be transformed in the same way that other functions can by stretching or compressing, reflecting, and shifting horizontally or vertically.

Recall the form of ordered pairs of input and output for any function: [latex]\left(x, f(x)\right)[/latex]. Using this well-defined relationship along with the fact that any point of form  [latex]\left(x, f(x)\right)[/latex] contained on the graph of a function satisfies the equation of the function (makes a true statement when the values of [latex]x[/latex] and [latex]f(x)[/latex] are substituted into the equation). That is, given a value for an input, we can substitute it for [latex]x[/latex] in the equation of a function to solve for the corresponding output. Likewise, given an value for an output, we can substitute it for [latex]f(x)[/latex] in the equation to solve for the corresponding input. And, given a point on the graph and an equation containing an unknown such as a leading coefficient or vertical intercept, we can substitute both input and output to solve for the unknown.

Recall the steps to evaluate and solve functions then practice these skills using the problems below. This will help you to prepare for handling the new exponential and logarithmic functions in the same way in this module.


  1. Replace the input variable in the formula with the value provided.
  2. Calculate the result.

How To: Solve a Function.

  1. Replace the output in the formula with the value provided.
  2. Solve for the input variable that makes the statement true.

Try It