Algebraic Functions

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ACT Math › Algebraic Functions

Questions 1 - 10

What is the domain of the function defined as follows:
$f(t) = \frac{2t}{4-t^2}$

$t = [-\infty,-2)\cup (-2,2)\cup (2,\infty]$

$\mathbb{R}$

$\mathbb{N}$

$[-\infty,0)\cup(0,\infty]$

$[-\infty,2)\cup(2,\infty]$

Explanation

The domain is the set of all where the function is defined. Since we can't divide by 0, we need to find the values that would make the denominator zero. Set the denominator equal to zero and solve for to find:

$\4-t^2 = 0 \newline 4 = t^2 \newline -2,2 = t$

Thus can take all values except for positive and negative 2, so the domain is:

$t = [-\infty,-2)\cup (-2,2)\cup (2,\infty]$

What is the domain of the function defined as follows:
$f(t) = \frac{2t}{4-t^2}$

$t = [-\infty,-2)\cup (-2,2)\cup (2,\infty]$

$\mathbb{R}$

$\mathbb{N}$

$[-\infty,0)\cup(0,\infty]$

$[-\infty,2)\cup(2,\infty]$

Explanation

$\4-t^2 = 0 \newline 4 = t^2 \newline -2,2 = t$

Thus can take all values except for positive and negative 2, so the domain is:

$t = [-\infty,-2)\cup (-2,2)\cup (2,\infty]$

An outpost has the supplies to last 2 people for 14 days. How many days will the supplies last for 7 people?

$\dpi{100} \small 4$

$\dpi{100} \small 7$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

$\dpi{100} \small 9$

Explanation

Supplies are used at the rate of $\dpi{100} \small \frac{Supplies}{Days\times People}$ .

Since the total amount of supplies is the same in either case, $\dpi{100} \small \frac{1}{14\times 2}=\frac{1}{7\times \ (&hash;\ of\ days)}$ .

Solve for days to find that the supplies will last for 4 days.

An outpost has the supplies to last 2 people for 14 days. How many days will the supplies last for 7 people?

$\dpi{100} \small 4$

$\dpi{100} \small 7$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

$\dpi{100} \small 9$

Explanation

Supplies are used at the rate of $\dpi{100} \small \frac{Supplies}{Days\times People}$ .

Since the total amount of supplies is the same in either case, $\dpi{100} \small \frac{1}{14\times 2}=\frac{1}{7\times \ (&hash;\ of\ days)}$ .

Solve for days to find that the supplies will last for 4 days.

When written in symbols, “The square of the sum of and equals ” is represented as:

Explanation

“The square of the sum” means that the summation of the terms is done first, and that summation is squared, which corresponds to the term .

When written in symbols, “The square of the sum of and equals ” is represented as:

Explanation

“The square of the sum” means that the summation of the terms is done first, and that summation is squared, which corresponds to the term .

Which of the following represents the domain of the function $f(x) = \sqrt{3x + 12}$ ?

$x \ge -4$

$x \ge 4$

$x \le 12$

$x \ge -12$

$x \ne 6$

Explanation

You know that the square root function has real roots only for positive values. Therefore, you know that $3x + 12 \ge 0$ . From this, you can solve for your domain:

$3x \ge -12$

$x \ge -4$

Which of the following represents the domain of the function $f(x) = \sqrt{3x + 12}$ ?

$x \ge -4$

$x \ge 4$

$x \le 12$

$x \ge -12$

$x \ne 6$

Explanation

You know that the square root function has real roots only for positive values. Therefore, you know that $3x + 12 \ge 0$ . From this, you can solve for your domain:

$3x \ge -12$

$x \ge -4$

A book binding company charges a fixed fee of \$2.25 to bind a book and an additional \$0.15 per page. Which equation accurately calculates the cost, C, of a book with p number of pages?

Explanation

The company's binding process incorporates a fixed fee; therefore, we must use the formula for a linear equation:

The fixed fee means that the consumer pays a single fee of \$2.25 to bind the book regardless of how many pages that book has; thus, the fixed fee is represented by the y-intercept, or b, of the equation. The problem states that each page will cost an additional \$0.15 per page, which varies with depending on the number of pages in the book. The page cost is represented by the slope, m, of the equation. In order to calculate the total cost, , you must multiply the number of pages, p, by \$0.15 and add the fixed cost of \$2.25; therefore, the following equation accurately models the cost of binding a book:

A book binding company charges a fixed fee of \$2.25 to bind a book and an additional \$0.15 per page. Which equation accurately calculates the cost, C, of a book with p number of pages?

Explanation

The company's binding process incorporates a fixed fee; therefore, we must use the formula for a linear equation:

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