Exercises

Find the weighted sum and the total weight separately.

The weighted sum is:

The total weight is $1+2+1=4$, so the weighted average is $20/4=5$. Dividing by the number of values instead would ignore the unequal weights.

Subtract $32$, then multiply by $5/9$.

Rearrange the formula:

Then:

Substituting $C=20$ into the original rule returns $F=68$, so the rearranged formula and the numerical result agree with the given temperature.

Evaluate $5^2$ and $2^5$ separately.

The outputs are:

Therefore $g(5)-f(5)=32-25=7$. One input does not establish which function grows faster everywhere, but it provides a concrete comparison.

Which exponent on $2$ produces $32$?

Taking the base-2 logarithm gives:

The check is $2^5=32$.

The expression gives a center of $1.5$ and a maximum distance of $0.2$.

The lower endpoint is $1.5-0.2=1.3$ and the upper endpoint is $1.5+0.2=1.7$. Equality is allowed, so:

Compute the intermediate value $h=f(2)$ first.

The first stage gives:

The second gives:

Naming the intermediate value makes the composition easier to inspect.

$\log_b(0)=x$ would require $b^x=0$.

For a positive base $b$, the value $b^x$ is positive for every finite real $x$. No finite exponent produces zero, so $\log_b(0)$ is not defined as a finite real number.

Only nonnegative inputs are allowed.

On all real numbers, $x^2$ maps $x$ and $-x$ to the same output. Restricting the domain to $x\ge0$ removes the negative input, so every nonnegative output has one allowed input:

Compute $a_1$, $a_2$, $a_3$, and $a_4$.

The sequence is:

Reaching $a_4$ from $a_0$ requires four updates. Equivalently, $a_n=16(1/2)^n$, so $a_4=16(1/2)^4=1$.

The original expression requires $x\ne1$. For allowed values of $x$, multiply both sides by $x-1$.

The denominator requires $x\ne1$. Multiplying by $x-1$ for allowed values gives:

Therefore $x=4$. This value is not excluded, and substitution confirms:

The symbol $\sqrt{49}$ names one principal value. The equation $x^2=49$ asks which real values square to $49$.

By definition, the principal square root is nonnegative, so $\sqrt{49}=7$. Both $7^2$ and $(-7)^2$ equal $49$, so:

The minimum occurs when $x-3=0$. Then evaluate $h(x)$.

The squared term is smallest when $x-3=0$, so $x=3$. At that input:

Therefore the vertex is $(3,2)$. The graph has moved three units right and two units up.

Composition gives $f(g(x))=\sqrt{x-5}$. A real square root requires its input to be nonnegative.

The output of $g$ becomes the input of $f$. Therefore:

The composition is defined for $x\in[5,\infty)$.

The first term is $2$, the ratio is $1/2$, and there are four terms.

With a common denominator:

This is a finite sum. It is less than the corresponding infinite sum, which approaches $4$.

Divide $240$ records by $6$ seconds, then use that rate for $1000$ records.

The processing rate is:

Therefore the required time is:

The record units cancel, leaving seconds.

First subtract $2$ from both sides. When you divide by $-3$, reverse the inequality.

Subtracting $2$ gives:

Dividing by $-3$ reverses the inequality:

As a check, $x=0$ satisfies both the solution set and the original inequality.

Solve $L=0.25L+6$. Then ask what that equation does and does not establish.

A fixed point remains unchanged:

Thus $8$ is a fixed point and a possible limit. The fixed-point equation alone does not prove that every initial value converges to it.

A negative ratio changes the sign. A ratio with magnitude less than $1$ reduces the magnitude.

The first terms are:

The factor $-1$ alternates the sign, while repeated multiplication by $1/2$ makes the magnitude approach $0$. Therefore $b_n$ converges to $0$ without becoming exactly zero at a finite index.