Lesson 16.5: Problem Solving with Functions

Property

To evaluate a function $f(a)$ when it is defined in the form $f(g(x)) = h(x)$, you must first solve the equation $g(x) = a$ to find the required value of $x$. Then, substitute this value of $x$ into the expression $h(x)$ to find the result.

Examples

• If $f(x - 3) = 2x + 1$, to find $f(5)$, we first solve $x - 3 = 5$, which gives $x = 8$. We then substitute $x=8$ into the expression $2x+1$ to get $2(8) + 1 = 17$. Thus, $f(5) = 17$.
• If $g(2x + 1) = x^2 - 4$, to find $g(7)$, we solve $2x + 1 = 7$, which gives $x = 3$. We then substitute $x=3$ into $x^2 - 4$ to get $3^2 - 4 = 5$. Thus, $g(7) = 5$.

Explanation

When a function''s input is an expression (like $x-3$) instead of just $x$, you cannot simply replace $x$ with the desired value. You must determine which value of $x$ makes the entire input expression equal to your target value. This process involves setting up and solving a small equation. Once you find the correct value for $x$, you substitute it into the output side of the function''s definition to get the final answer.

Property

To solve an equation involving a function with a transformed argument, such as finding $z$ where $f(h(z)) = C$, given the definition of $f(g(x))$, first find the necessary substitution. Set the arguments equal, $g(x) = h(z)$, and solve for $x$ in terms of $z$. Substitute this expression for $x$ into the definition of $f(g(x))$ to get an expression for $f(h(z))$, then solve for $z$.

Examples

Example 1

Given $f(x+2) = x^2 - 1$, find the sum of all values of $z$ for which $f(2z) = 3$.
Let $x+2 = 2z$, so $x = 2z - 2$.
Substitute $x$ into the function definition: $f(2z) = (2z - 2)^2 - 1 = 3$.
$(2z - 2)^2 = 4$, so $2z - 2 = 2$ or $2z - 2 = -2$.
This gives $2z = 4$ or $2z = 0$, so $z=2$ or $z=0$. The sum is $2+0=2$.

Example 2

If $f(\frac{x}{4}) = x - 5$, find the value of $z$ such that $f(z+1) = 15$.
Let $\frac{x}{4} = z+1$, so $x = 4(z+1) = 4z+4$.
Substitute $x$: $f(z+1) = (4z+4) - 5 = 15$.
$4z - 1 = 15$, so $4z = 16$, which means $z=4$.

Explanation

This skill involves a two-step substitution process to solve equations with functions. First, you relate the known function input to the desired function input by setting them equal. Solving this equation gives you the required substitution to transform the function. Finally, you apply this substitution to the function''s definition, creating a new equation that you can solve for the unknown variable.

Property

To compare the outputs of two functions, $f(x)$ and $g(x)$, based on a real-world condition, we can set up an inequality. For instance, if the value of $f(x)$ must be less than the value of $g(x)$, the condition is represented by the inequality $f(x) < g(x)$.

Examples

• Two companies’ profits are modeled by the functions $P_1(t) = 5000 + 300t$ and $P_2(t) = 8000 + 150t$, where $t$ is months. To find when Company 1’s profit exceeds Company 2’s, we solve the inequality $P_1(t) > P_2(t)$.
• The populations of two towns are given by $A(y) = 1200(1.03)^y$ and $B(y) = 1500(1.01)^y$, where $y$ is the number of years. To find when the population of Town A is at least that of Town B, we solve $A(y) \geq B(y)$.

Explanation

This skill involves translating a word problem into a mathematical model using two or more functions. A condition comparing the outcomes of these functions (e.g., one being greater, smaller, or equal to another) is then expressed as an inequality. Solving this inequality for the input variable provides the solution to the original problem. This approach is common in economics, biology, and other fields for comparing different growth or decay models.

Property

A function can be defined by a functional equation that describes relationships between function values, such as $f(a) + f(b) = f(ab)$ or $f(x + y) = f(x) + f(y)$, without giving an explicit formula for $f(x)$.

Examples