# 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 2

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

Substitute '7' in for x :

### 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

### Practice Problems

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

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

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

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

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

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

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

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

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

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

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

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

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 $$(\color{blue} {t}, \color{red} {h(t)}) (\color{blue} {4}, \color{red} {81.2})$$ :
Graph generated by Meta Calculator's graphing calc

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 $$(\color{blue} {t}, \color{red} {h(t)}) (\color{blue} {98} , \color{red} {16.5159})$$ :

Graph generated by Meta Calculator's graphing calc