Functions Practice

No. A function must assign each input exactly one output.

Here the input $3$ is paired with two different outputs, $7$ and $9$, so the relation is not a function.

The denominator cannot be zero, so solve

which gives

So the domain excludes $x = 3$.

Substitute $4$ for $x$:

Evaluate:

Look at the row where $x = 3$.

The corresponding output is

The expression inside the square root must be nonnegative:

Solve for $x$:

So the domain is all real numbers less than or equal to $8$.

Substitute $0$ for $x$:

Any nonzero number raised to the $0$ power equals $1$, so

No. A function must be one-to-one to have an inverse on its full domain.

Here two different inputs, $1$ and $2$, produce the same output, $4$, so the function is not one-to-one.

Since $-3 < 0$, use the first rule:

Squares are never negative, so every output satisfies

Also, every nonnegative number occurs as a square of some real number. Therefore the range is

The argument of a logarithm must be positive:

So

By definition,

Substitute $g(x) = 3x - 2$ into $f$:

Expand:

Write

Swap $x$ and $y$:

Solve for $y$:

So

The graph of $x^2$ has minimum value $0$.

Shifting it down by $6$ moves the minimum to $-6$.

So the range is

Factor the numerator:

So for $x \ne 4$,

Now substitute $x = 7$:

Use the average rate of change formula:

Compute the function values:

Then

For continuity at $x = 2$, the left and right values must match.

The right-hand value at $x = 2$ is

So the left-hand expression must also equal $7$ at $x = 2$:

The leading term is $-2x^5$, so it determines the end behavior.

As $x \to \infty$,

As $x \to -\infty$,

Read the expression from the inside out.

moves the graph left $3$ units.

The factor $-2$ reflects the graph across the $x$-axis and stretches it vertically by a factor of $2$.

The $+1$ shifts the graph up $1$ unit.

So the transformations are:

• left $3$

• vertical stretch by $2$

• reflect across the $x$-axis

• up $1$

Let $l$ be the number of lessons. Set up the equation:

Subtract $25$:

Divide by $18$:

Use the average rate of change formula:

Compute the values:

So

The average rate of change is $10$ centimeters per week.

Doubling every hour means the model is exponential:

Now evaluate at $t = 5$:

After the first hour, there are $4$ additional hours.

The total cost is

So the cost is $\$16$.

Set $C(F) = 20$:

Multiply both sides by $\frac{9}{5}$:

Add $32$:

So the temperature is $68^\circ\text{F}$.

Write

Swap $x$ and $y$:

Because the original domain is $y \ge 2$, choose the positive square root:

So

The domain of the inverse is $x \ge 0$.

The composition is

Two conditions must hold:

1. $x - 1 \ge 0$ so that the square root is defined.

2. $\sqrt{x - 1} > 0$ because the input of a logarithm must be positive.

The second condition means

so the domain is

Factor the numerator:

So for $x \ne 3$,

This is one-to-one, so the function does have an inverse on its domain.

To find it, write

and swap variables:

Solve for $y$:

So

Since the original range excludes $6$, the domain of the inverse is $x \ne 6$.

Use the definitions of even and odd functions.

For $f \circ g$:

because $g$ is odd and $f$ is even. So $f \circ g$ is even.

For $g \circ f$:

because $f$ is even, so the input to $g$ does not change when $x$ is replaced by $-x$. Thus $g \circ f$ is also even.