Evaluate Functions

How do you evaluate functions?

The same way that you substitute values into equations!

For instance, consider the following problem,

Example 1

What is the value of $$ x $$ given the equation $$ y = 2x $$ when $$ x = 5 $$?

Substitute '5' in for x :

The one new aspect of function notation is the emphasis on input and output

example of how to evaluate a function in math Example 2

What is the value of $$ x $$ given the equation $$ y = x-5 $$ when $$ x = 7 $$?

Substitute '7' in for x :

Again, this new way involves an input and output

example 2 of how to evaluate a function

Function Machine

You can think of $$ f(x)$$ as a function machine. The function machine, or $$ f(x)$$, takes input inside. The machine processes this input and produces an output value

function machine-math functions evaluate

Practice Problems

Problem 1

Let $$ k(x) =3x $$

Evaluate $$ k (5 ) $$

function machine question
Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ k( \color{blue}{x }) =3 \color{blue}{x } \\ k( \color{blue}{5 }) =3\cdot \color{blue}{5 } $$

Step 2

compute result

$$ k( \color{blue}{5 }) =3\cdot \color{blue}{5 } \\ = \color{red}{15} $$


function machine in 5 out 15

$$ k(\color{blue} {input } ) =\color{red} {output} \\ = k(\color{blue} {5} ) =\color{red} {15} $$

Problem 2

Let $$ g(x) =3x^2 + 7x $$.

Evaluate $$ g( 4) $$

function machine g of x input is 5
Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ g( \color{blue}{x }) =3 \color{blue}{x }^2 + 7\color{blue}{x } \\ g( \color{blue}{4 }) =3\cdot \color{blue}{4 }^2 + 7\cdot \color{blue}{4 } $$

Step 2

compute result

$$ g( \color{blue}{4 }) =3\cdot \color{blue}{4 }^2 + 7\cdot \color{blue}{4 } \\ g( \color{blue}{4 }) = \color{red}{76} $$


function machine in 5 out 15

$$ g(\color{blue} {input } ) =\color{red} {output} \\ g(\color{blue} {4} ) =\color{red} {76} $$

Problem 3

Let $$ h(x) =\sqrt{x^3 -4}-|x| $$

Evaluate $$ h( 5) $$

Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ h( \color{blue}{x }) = \sqrt{\color{blue}{x }^3 -4}-|\color{blue}{x }| \\ h( \color{blue}{5 }) = \sqrt{\color{blue}{5 }^3 -4}-|\color{blue}{5 }| $$

Step 2

compute result

$$ h( \color{blue}{5 }) = \sqrt{\color{blue}{5 }^3 -4}-|\color{blue}{5 }|=50 \\ h( \color{blue}{ 5 }) = \color{red}{ 6 } $$


$$ h(\color{blue} {input } ) =\color{red} {output} \\ h(\color{blue} {5} ) =\color{red} { 6 } $$

Problem 4

Let $$ f(x) = -3x^2 + 5x - 1 $$

Evaluate $$ f(6) $$

Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ f( \color{blue}{x }) = -3 \color{blue}{x} ^2 + 5\color{blue}{x} - 1 \\ f( \color{blue}{ 6 }) = -3 \cdot \color{blue}{6} ^2 + 5\cdot \color{blue}{6} - 1 $$

Step 2

compute result

$$ f( \color{blue}{ 6 }) = -3 \cdot \color{blue}{6} ^2 + 5\cdot \color{blue}{6} - 1 \\ f( \color{blue}{6}) = \color{red}{–79} $$


$$ f(\color{blue} {input } ) =\color{red} {output} \\ f(\color{blue} {6} ) =\color{red} {–79} $$

Problem 5

The height in meters of a projectile at t seconds can be found by the function $$ h(t) = -5t^2 + 40t + 1.2 $$.

Find the height of the projectile 4 seconds after it is launched.

Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ h( \color{blue}{t }) = -5\color{blue}{t}^2 + 40 \color{blue}{t} + 1.2 \\ h( \color{blue}{4 }) = -5 \cdot \color{blue}{4}^2 + 40 \cdot \color{blue}{4} + 1.2 $$

Step 2

compute result

$$ h( \color{blue}{4 }) = -5 \cdot \color{blue}{4}^2 + 40 \cdot \color{blue}{4} + 1.2 \\ h( \color{blue}{4}) = \color{red}{81.2} $$


$$ h(\color{blue} {input } ) =\color{red} {output} \\ h(\color{blue} {4} ) =\color{red} {81.2} $$

Here is a picture of graph of projectile's path with the point $$ (\color{blue} {t}, \color{red} {h(t)}) (\color{blue} {4}, \color{red} {81.2}) $$ :
real world evaluate function Graph generated by Meta Calculator's graphing calc

Problem 6

A substance has a half life of 26 years. The amount of remaining substance in grams after t years can be found by the function $$ h(t) = 250 (0.5)^{ \frac{t}{25} } $$.

How much substance remains after 98 years?

Step 1

Identify all of the occurrences of 'x' and substitute the input in

$$ h(\color{blue}{t}) = 250 (0.5)^{ \frac{\color{blue}{t}}{25} } \\ h(\color{blue}{98}) = 250 (0.5)^{ \frac{\color{blue}{98}}{25} } $$

Step 2

compute result

$$ h(\color{blue}{98}) = 250 (0.5)^{ \frac{\color{blue}{98}}{25} } \\ h( \color{blue}{98}) = \color{red}{16.5159} $$


$$ h(\color{blue} {input } ) =\color{red} {output} \\ h(\color{blue} {98} ) =\color{red} {16.5159} $$

Here is a picture of graph of projectile's path with the point $$ (\color{blue} {t}, \color{red} {h(t)}) (\color{blue} {98} , \color{red} {16.5159}) $$ :

real world evaluate function Graph generated by Meta Calculator's graphing calc
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