01 Functions and Their Graphs

Understanding Functions and Notation

A function connects each allowed input to exactly one output. The input set is the domain, and the resulting output set is the range. Functions can be represented in several equivalent ways:

• Verbally: a description such as “the cost depends on the number of items.”

• Numerically: a table of input-output pairs.

• Algebraically: a rule such as $f(x)=2x+3$.

• Graphically: points or a curve in the coordinate plane.

In function notation, $f(x)$ means the output produced by the input $x$. For example, if $f(x)=2x+3$, then

To evaluate a formula, substitute the input everywhere the variable occurs. If $f(x)=x^2-4x+1$, then

A function can also be used in reverse to find inputs that produce a specified output. For $f(x)=x^2-4$, solving $f(x)=12$ gives

Thus, two different inputs can produce the same output without violating the definition of a function.

Takeaway: Read $f(x)$ as an output rule, evaluate by substitution, and remember that uniqueness applies from input to output.

Domains, Ranges, and Graphs

The domain is the set of permitted inputs, while the range is the set of outputs generated by those inputs. When a function is given by a formula, start with all real numbers and remove inputs that make the formula undefined.

Important restrictions include:

• A denominator cannot equal zero. For $f(x)=\frac{1}{x-5}$, require $x-5\ne0$, so the domain is $(- \infty,5)\cup(5,\infty)$.

• An even root requires a nonnegative radicand. For $g(x)=\sqrt{x+2}$, require $x+2\ge0$, so the domain is $[-2,\infty)$.

• A logarithm requires a positive input. For $h(x)=\log(x-1)$, require $x-1>0$, so the domain is $(1,\infty)$.

In interval notation, parentheses exclude endpoints and brackets include them. Infinity is never included, so it always appears with a parenthesis. From a graph, the domain describes the horizontal extent and the range describes the vertical extent.

The graph of a function consists of all points $(x,f(x))$ for inputs in the domain. This viewpoint connects formulas and graphs: the horizontal coordinate is the input, and the vertical coordinate is the corresponding output.

Takeaway: Check algebraic restrictions before evaluating or graphing, and use the graph's horizontal and vertical extents to identify domain and range.

Reading and Interpreting Function Graphs

A graph represents a function when each input has no more than one output. The vertical line test provides a quick graphical check: if any vertical line crosses the graph more than once, the graph is not a function of $x$. A parabola such as $y=x^2$ passes this test, while a circle does not.

To graph a formula, identify its domain, choose useful inputs, calculate outputs, plot the ordered pairs, and connect them according to the function's behavior. For $f(x)=x^2$, representative points include $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, and $(2,4)$. The graph is a parabola with vertex $(0,0)$ and symmetry about the $y$-axis because $f(-x)=f(x)$.

Intercepts give important reference points:

• The $y$-intercept occurs when $x=0$, provided zero is in the domain.

• An $x$-intercept occurs when the output is zero, so solve $f(x)=0$.

For $f(x)=x^2-4$, the $y$-intercept is $(0,-4)$. Solving

gives $x=2$ or $x=-2$, so the $x$-intercepts are $(2,0)$ and $(-2,0)$.

A one-to-one function has a further property: different inputs always produce different outputs. Its graph passes the horizontal line test. For example, $f(x)=x^3$ is one-to-one on the real numbers, but $f(x)=x^2$ is not because $f(-2)=f(2)=4$.

Takeaway: Use the vertical line test to identify functions, the horizontal line test to identify one-to-one functions, and intercepts and symmetry to interpret graphs efficiently.

Transforming Parent Functions

Many graphs are built from a parent function by shifting, reflecting, stretching, or compressing it. A useful general form is

The parameters control different changes:

• $h$ controls the horizontal shift.

• $k$ controls the vertical shift.

• $a$ controls vertical stretch or compression and may reflect the graph across the $x$-axis.

• $b$ controls horizontal compression or stretch and may reflect the graph across the $y$-axis.

For vertical changes, $g(x)=f(x)+k$ shifts the graph up when $k>0$ and down when $k<0$. Multiplying by $a$ stretches vertically when $|a|>1$, compresses vertically when $0<|a|<1$, and reflects across the $x$-axis when $a<0$.

For horizontal changes, $g(x)=f(x-h)$ shifts the graph right by $h$ when $h>0$. The sign can seem reversed because the change occurs inside the function. Thus, $g(x)=(x-3)^2$ is $y=x^2$ shifted right by $3$ units. Also, $f(-x)$ reflects across the $y$-axis.

Consider

Starting with $f(x)=x^2$, shift right by $1$, stretch vertically by a factor of $2$, reflect across the $x$-axis, and shift up by $4$. The vertex is $(1,4)$, and the parabola opens downward.

Takeaway: Read transformations from the outside and inside of the formula, paying special attention to the sign of horizontal changes.

Compositions and Inverse Functions

The composition of functions uses the output of one function as the input of another. It is written

The function on the inside, $g$, is applied first. For $f(x)=2x+1$ and $g(x)=x^2$,

Reversing the order gives

which is generally different. To evaluate $(f\circ g)(3)$, first find $g(3)=9$, then find $f(9)=19$. Therefore, $(f\circ g)(3)=19$.

The domain of a composition must satisfy both stages: the input must belong to the domain of $g$, and the output $g(x)$ must belong to the domain of $f$. If $f(x)=\sqrt{x}$ and $g(x)=x-3$, then

The requirement $x-3\ge0$ gives the domain $[3,\infty)$.

An inverse function reverses the input-output process. It exists as a function only when the original function is one-to-one on its domain. To find the inverse of $f(x)=3x-5$, write $y=3x-5$, solve for $x$, and interchange the variables:

The result can be checked through

The domain of an inverse is the range of the original function, and the range of the inverse is the original domain. Their graphs reflect across $y=x$. If $f(x)=x^2$, an inverse function requires a restricted domain such as $[0,\infty)$; on that domain, the inverse is $f^{-1}(x)=\sqrt{x}$.

Takeaway: In compositions, work from the inside out; for inverses, verify one-to-one behavior and exchange the original domain and range.