Use the Graph Shown to Evaluate the Composition Es001-1jpg

Use the Graph Shown to Evaluate the Composition Es001-1jpg.

Once we etch a new part from 2 existing functions, we need to exist able to evaluate it for any input in its domain. We volition exercise this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables every bit inputs to functions expressed every bit formulas. In each case we evaluate the inner function using the starting input and and so use the inner office’s output as the input for the outer role.

Evaluating Composite Functions Using Tables

When working with functions given as tables, nosotros read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside part first and and so use the output of the inside office every bit the input to the outside function.

Example: Using a Table to Evaluate a Composite Part

Using the table below, evaluate [latex]f\left(1000\left(3\correct)\right)[/latex] and [latex]chiliad\left(f\left(three\right)\right)[/latex].

[latex]x[/latex] [latex]f\left(x\right)[/latex] [latex]g\left(10\right)[/latex]
one 6 three
two 8 5
three 3 2
iv 1 vii

Try It

Using the table beneath, evaluate [latex]f\left(g\left(i\right)\right)[/latex] and [latex]m\left(f\left(4\right)\right)[/latex].

[latex]10[/latex] [latex]f\left(10\right)[/latex] [latex]g\left(x\right)[/latex]
one 6 3
ii 8 5
3 iii 2
4 i 7

[latex]f\left(g\left(1\correct)\right)=f\left(three\correct)=3[/latex] and [latex]g\left(f\left(4\right)\correct)=g\left(i\right)=3[/latex]

Evaluating Composite Functions Using Graphs

When we are given individual functions every bit graphs, the process for evaluating composite functions is similar to the procedure we use for evaluating tables. We read the input and output values, merely this time, from the [latex]x\text{-}[/latex] and [latex]y\text{-}[/latex] axes of the graphs.

How To: Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.

  1. Locate the given input to the inner function on the [latex]x\text{-}[/latex] axis of its graph.
  2. Read off the output of the inner part from the [latex]y\text{-}[/latex] axis of its graph.
  3. Locate the inner function output on the [latex]x\text{-}[/latex] axis of the graph of the outer function.
  4. Read the output of the outer function from the [latex]y\text{-}[/latex] axis of its graph. This is the output of the composite part.

Example: Using a Graph to Evaluate a Composite Office

Using the graphs below, evaluate [latex]f\left(g\left(1\right)\right)[/latex].

Explanation of the composite function.

Attempt Information technology

Using the graphs beneath, evaluate [latex]g\left(f\left(2\right)\right)[/latex].

Two graphs of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.

[latex]thousand\left(f\left(2\right)\right)=one thousand\left(5\right)=3[/latex]

Evaluating Composite Functions Using Formulas

When evaluating a blended function where we accept either created or been given formulas, the rule of working from the within out remains the same. The input value to the outer part will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, information technology is sometimes helpful to find a single formula that will calculate the outcome of a composition [latex]f\left(g\left(x\right)\correct)[/latex]. To do this, we volition extend our idea of role evaluation. Think that, when we evaluate a function similar [latex]f\left(t\correct)={t}^{2}-t[/latex], we substitute the value inside the parentheses into the formula wherever we see the input variable.

How To: Given a formula for a composite function, evaluate the function.

  1. Evaluate the inside function using the input value or variable provided.
  2. Use the resulting output as the input to the exterior function.

Example: Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input

Given [latex]f\left(t\correct)={t}^{2}-{t}[/latex] and [latex]h\left(10\right)=3x+ii[/latex], evaluate [latex]f\left(h\left(ane\right)\correct)[/latex].

Try Information technology

Given [latex]f\left(t\right)={t}^{2}-t[/latex] and [latex]h\left(x\correct)=3x+ii[/latex], evaluate

A) [latex]h\left(f\left(2\right)\right)[/latex]

B) [latex]h\left(f\left(-ii\right)\right)[/latex]


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Use the Graph Shown to Evaluate the Composition Es001-1jpg