Substituting Expressions into Function Notation

Introduction

Function notation can contain an input that is more complicated than a single number. For example, \(f(a+h)\) asks for the output of \(f\) when its input is the whole expression \(a+h\). The function is not being multiplied by \(a+h\), and the symbols \(a\) and \(h\) do not replace the function’s rule.

The central operation is substitution: take the formula that defines the function and replace every occurrence of its input variable with the entire new input. Parentheses help keep that input together while simplifying.

Key idea: In \(f(u)\), the expression \(u\) is one input to the function. If the rule for \(f\) is given by a formula, put \(u\) in place of every occurrence of the formula’s input variable.

The Input Is the Entire Expression

Suppose a function is defined by \(f(x)=x^2+3x\). The letter \(x\) is a placeholder for the input. The rule says: square the input, then add three times the input. If the input is \(a+h\), both occurrences of \(x\) in the rule must be replaced by \(a+h\): the input is squared as a whole, and the input is also multiplied by \(3\).

$$ f(a+h)=(a+h)^2+3(a+h). $$

It would not be correct to replace only one occurrence of \(x\), or to replace \(x\) by \(a\) and then add \(h\) to the final output. The formula describes what to do with one input; each appearance of the input variable refers to that same input.

Definition: If a function has a formula \(f(x)=F(x)\), then evaluating it at an input expression \(u\) means forming \(f(u)=F(u)\), with \(u\) substituted for every occurrence of \(x\) in \(F(x)\). The input \(u\) must be an allowed input for the function.

When the new input contains addition or subtraction, put it in parentheses wherever it replaces the variable. This makes clear which parts belong to the input before applying powers or other operations. For instance, replacing \(x\) by \(2x-1\) in \(x^2\) gives \((2x-1)^2\), not \(2x-1^2\).

A Reliable Substitution Method

A short sequence of steps keeps the input and the formula distinct. First identify the full input written inside the function notation. Next replace every occurrence of the formula’s input variable with that full expression, using parentheses where needed. Finally simplify, preserving the original order of operations.

Identify the input.
In \(f(2x-1)\), the input is the complete expression \(2x-1\).

Replace every input variable.
In the formula for \(f\), replace each \(x\) with \((2x-1)\).

Simplify and check.
Expand or combine terms as appropriate, then test the result at a convenient value if possible.

This method works because a function’s formula assigns an output to each allowed input. Once an input expression \(u\) is chosen, the formula \(F\) is evaluated at that value: \(f(u)=F(u)\). If \(x\) appears several times in \(F(x)\), every occurrence represents the same input, so each must become \(u\). Substitution does not change the rule; it applies the same rule to a different input.

Worked Examples

Worked Example: Finding \(f(a+h)\)

Let \(f(x)=x^2-4x+1\). Find and simplify \(f(a+h)\).

The complete input is \(a+h\). Replace each \(x\) in the formula by \((a+h)\):

\begin{aligned} f(a+h) &=(a+h)^2-4(a+h)+1\\ &=(a^2+2ah+h^2)-4a-4h+1\\ &=a^2+2ah+h^2-4a-4h+1. \end{aligned}

For a check, choose \(a=2\) and \(h=3\). The input is \(a+h=5\). Evaluating the original rule gives \(f(5)=5^2-4(5)+1=25-20+1=6\). Substituting those values into the simplified expression gives \(2^2+2(2)(3)+3^2-4(2)-4(3)+1=4+12+9-8-12+1=6\). The two evaluations agree.

Worked Example: Finding \(g(2x-1)\)

Let \(g(x)=2x^2+3x-1\). Find and simplify \(g(2x-1)\).

The input \(2x-1\) replaces each \(x\), including the \(x\) that is squared and the \(x\) in the linear term:

\begin{aligned} g(2x-1) &=2(2x-1)^2+3(2x-1)-1\\ &=2(4x^2-4x+1)+6x-3-1\\ &=8x^2-8x+2+6x-4\\ &=8x^2-2x-2. \end{aligned}

Check the result at \(x=2\). The input to \(g\) is \(2(2)-1=3\), so the original rule gives \(g(3)=2(3^2)+3(3)-1=18+9-1=26\). The simplified expression gives \(8(2^2)-2(2)-2=32-4-2=26\). Thus the substitution and simplification give the same output.

Worked Example: Substituting into a Rule with One Input Term

Let \(p(t)=5t-2\). Find \(p(3y+4)\).

The formula uses \(t\) as its input variable, while the new input is \(3y+4\). Replace \(t\) with the whole expression \(3y+4\):

\begin{aligned} p(3y+4) &=5(3y+4)-2\\ &=15y+20-2\\ &=15y+18. \end{aligned}

Check at \(y=-1\). The input is \(3(-1)+4=1\), and the original rule gives \(p(1)=5(1)-2=3\). The simplified expression gives \(15(-1)+18=-15+18=3\). The values match.

Substitution Is Not Multiplication

The notation \(f(2x)\) means “the value of \(f\) at input \(2x\).” It does not mean \(2\) multiplied by \(f(x)\). These expressions can produce different results because they apply different operations.

For example, use \(g(x)=2x^2+3x-1\). Substituting \(2x\) for the input gives \(g(2x)=2(2x)^2+3(2x)-1=8x^2+6x-1\). Multiplying the output \(g(x)\) by \(2\) instead gives \(2g(x)=2(2x^2+3x-1)=4x^2+6x-2\). The resulting formulas are different.

Notation reminder: \(f(u)\) means evaluate the function at input \(u\). The expression \(c f(x)\) means multiply the output \(f(x)\) by \(c\). Do not treat the expression inside the parentheses as a factor multiplying the function.

Common Mistake: Forgetting Parentheses

A frequent error is to substitute correctly in some parts of a formula but then lose the grouping in a power. For \(g(x)=2x^2+3x-1\), the incorrect calculation \(g(2x-1)=2(2x-1^2)+3(2x-1)-1\) fails to square the entire input. In fact, \((2x-1)^2\) is \((2x-1)(2x-1)\), which expands to \(4x^2-4x+1\), not \(2x-1^2\).

The correction is to keep the input together during substitution and then apply the exponent to the whole parenthesized expression:

g(2x-1)=2\bigl((2x-1)^2\bigr)+3(2x-1)-1.

Only after that should the square be expanded. Parentheses are not decoration: they record which expression is the input and which operation applies to it.

How to Check a Substitution

A check at one convenient numerical value is useful for finding mistakes in an expanded result. Substitute the same value into both the original function notation and the proposed simplified expression. The inputs and outputs should agree. This check can reveal an algebra error, although agreement at one value alone does not prove that two formulas are equal for every value. The substitution steps and algebraic simplification establish the general result.

When checking, be especially careful to compute the new input first. For \(f(2x-1)\), evaluate \(2x-1\) and then use that result as the input to \(f\). Do not evaluate \(f(x)\) first and then try to adjust its output. The function rule acts on its input as a whole.

Summary

To find a function’s value at an expression, identify the complete input, replace every occurrence of the formula’s input variable with that expression, and simplify. Parentheses preserve the structure of inputs involving sums or differences, particularly when the input is raised to a power. Finally, a numerical check can help confirm the calculation.

Check Your Understanding

For each question, identify the full input before simplifying. Use a numerical check where requested.

If \(f(x)=x^2+2x\), write and simplify \(f(a+1)\).
If \(g(t)=3t-5\), find \(g(2y-4)\).
If \(p(x)=x^2-1\), write \(p(3x)\). Explain why it is not the same notation as \(3p(x)\).
If \(q(x)=x^2+4x\), find and simplify \(q(1-h)\).
For your answer to question 2, check the result at \(y=2\) by evaluating the input in the original function rule and in your simplified expression.

Substituting Expressions into Function Notation

What You'll Learn

Introduction

The Input Is the Entire Expression

A Reliable Substitution Method