PSAT Math - How to find f(x)

If f(x) = _x_2 – 5 for all values x and f(a) = 4, what is one possible value of a?

Explanation

Using the defined function, f(a) will produce the same result when substituted for x:

f(a) = _a_2 – 5

Setting this equal to 4, you can solve for a:

_a_2 – 5 = 4

_a_2 = 9

a = –3 or 3

Explanation

When we multiply a function by a constant, we multiply each value in the function by that constant. Thus, 2f(x) = 4x + 12. We then subtract g(x) from that function, making sure to distribute the negative sign throughout the function. Subtracting g(x) from 4x + 12 gives us 4x + 12 - (3x - 3) = 4x + 12 - 3x + 3 = x + 15. We then add 2 to x + 15, giving us our answer of x + 17.

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.

Let $f(x)$ and $g(x)$ be functions such that $f(x)=\frac{1}{x-3}$ , and $f(g(x))=g(f(x))=x$ . Which of the following is equal to $g(x)$ ?

$\frac{3x+1}{x}$

$\frac{1}{x+3}$

$\frac{x}{x-3}$

$\frac{4}{x}$

$\frac{x}{x+3}$

Explanation

If $\dpi{100} h(x)$ and $\dpi{100} k(x)$ are defined as inverse functions, then $\dpi{100} h(k(x))=k(h(x))=x$ . Thus, according to the definition of inverse functions, $\dpi{100} f(x)$ and $\dpi{100} g(x)$ given in the problem must be inverse functions.

If we want to find the inverse of a function, the most straighforward method is usually replacing $\dpi{100} f(x)$ with $\dpi{100} y$ , swapping $\dpi{100} y$ and $\dpi{100} x$ , and then solving for $\dpi{100} y$ .

We want to find the inverse of $f(x)=\frac{1}{x-3}$ . First, we will replace $\dpi{100} f(x)$ with $\dpi{100} y$ .

$y = \frac{1}{x-3}$

Next, we will swap $\dpi{100} x$ and $\dpi{100} y$ .

$x = \frac{1}{y-3}$

Lastly, we will solve for $\dpi{100} y$ . The equation that we obtain in terms of $\dpi{100} x$ will be in the inverse of $\dpi{100} f(x)$ , which equals $\dpi{100} g(x)$ .

We can treat $x = \frac{1}{y-3}$ as a proportion, $\frac{x}{1} = \frac{1}{y-3}$ . This allows us to cross multiply and set the results equal to one another.

$x(y-3)= 1$

We want to get y by itself, so let's divide both sides by x.

$y-3=\frac{1}{x}$

Next, we will add 3 to both sides.

$y=\frac{1}{x}+3$

To combine the right side, we will need to rewrite 3 so that it has a denominator of $\dpi{100} x$ .

$y=\frac{1}{x}+3 = \frac{1}{x}+\frac{3x}{x}=\frac{3x+1}{x}$

The answer is $\frac{3x+1}{x}$ .

The cost of a cell phone plan is $40 for the first 100 minutes of calls, and then 5 cents for each minute after. If the variable x is equal to the number of minutes used for calls in a month on that cell phone plan, what is the equation f(x) for the cost, in dollars, of the cell phone plan for calls during that month?

f(x) = 40 + 5x

f(x) = 40 + 0.5x

f(x) = 40 + 0.05x

f(x) = 40 + 0.5(x - 100)

f(x) = 40 + 0.05(x - 100)

Explanation

40 dollars is the constant cost of the cell phone plan, regardless of minute usage for calls. We then add 5 cents, or 0.05 dollars, for every minute of calls over 100. Thus, we do not multiply 0.05 by x, but rather by (x - 100), since the 5 cent charge only applies to minutes used that are over the 100-minute barrier. For example, if you used 101 minutes for calls during the month, you would only pay the 5 cents for that 101st minute, making your cost for calls $40.05. Thus, the answer is 40 + 0.05(x - 100).

f(x) = 4x + 2

g(x) = 3x - 1

The two equations above define the functions f(x) = g(x). If f(d) = 2g(d) for some value of d, then what is the value of d?

-2

1/2

3/2

Explanation

f(x) = 4x + 2

g(x) = 3x - 1

We have f(d) = 2g(d). We multiply each value in g(d) by 2.

4d + 2 = 2(3d - 1) (Distribute the 2 in the parentheses by multiplying each value in them by 2.)

4d + 2 = 6d - 2 (Add 2 to both sides.)

4d + 4 = 6d (Subtract 4d from both sides.)

4 = 2d (Divide both sides by 2.)

2 = d

We can plug that back in to double check.

4(2) + 2 = 6(2) - 2

8 + 2 = 12 - 2

10 = 10

The function f, where f(x) = x2 + 6x + 8, is related to function g, where g(x) = 5 f(x-2). What is g(3)?

125

150

175

Explanation

Doing things in an orderly way is a friend to the test-taker.

g(3) = 5 f(3-2)

= 5 f(1)

= 5 \[ 12 + 6**∙**1 + 8\]

= 5 \[ 1 + 6 + 8\]

= 5 \[ 15\]

= 75

Let the function f be defined by f(x)=x-t. If f(12)=4, what is the value of f(0.5*t)?

Explanation

First we substitute in 12 for x and set the equation up as 12-t=4. We then get t=8, and substitute that for t and get f(0.5*8), giving us f(4). Plugging 4 in for x, and using t=8 that we found before, gives us:

f(4) = 4 - 8 = -4

The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?

p+mn

p+m(n+1)

p+m(n-1)

pn+m(n-1)

Explanation

The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).

Let f(x) be a function with at least one root. All of the following graphs must have the exact same root(s) as f(x) EXCEPT:

(f(x))(1/2)

(f(x))2

–4_f_(x)

|f(x)|

(f(x))–1

Explanation

Remember that a root of a function is the point where it crosses the x-axis, i.e. an x-intercept. The x-intercepts of a function occur where the y-value of a point is equal to zero. Therefore, the solutions to the equation f(x) = 0 give us the roots of f(x). Thus, if we set the functions in the answer choices equal to 0, and we end up solving the equation f(x) = 0, then the function will have the same roots as f(x) .

Let's set each of the functions in the answer choices equal 0.

First, let's look at (f(x))(1/2) = 0. Raising a function to the 1/2 power is the same as taking the square root of f(x). To get rid of the square root, we can square both sides.

(f(x))(1/2) = √(f(x)) = 0

(√f(x))2 = 02

f(x) = 0. Solving the equation f(x) = 0 will give us the roots of f(x). This means, to find the roots of (f(x))(1/2), we will end up having to find the roots of f(x). In short, the roots of (f(x))(1/2) will be the same as those of f(x).

Intuitively, this makes sense. When we graph (f(x))(1/2) , what we are doing is taking the square root of the y-values of every point on f(x). The roots of f(x) will all have y-values of zero, and taking the square root of zero isn't going to change the y-value. Thus, the location of the roots on f(x) will not change, and the roots of (f(x))(1/2) will be the same as the roots of f(x).

Next, we can look at (f(x))2. We can set it equal to zero to find the roots.

(f(x))2 = 0

We can then take the square root of both sides. The square root of zero is zero.

f(x) = 0

Once again, we are left with the equation f(x) = 0, which will give us the roots of f(x). This makes sense, because (f(x))2 means we are squaring the y-value of the roots on f(x), which wouldn't change the location of the roots, because the square of zero is still zero.

The third graph is –4_f_(x). Once again, we can set it equal to zero.

–4_f_(x) = 0

Divide both sides by –4.

f(x) = 0

We are left with this equation, which will give us the roots of f(x). This makes sense, because if we were to multiply the y-values of the x-intercepts on f(x) by –4, then the location of the points wouldn't change. Multiply 0 by –4 will still give us 0.

The next function is |f(x)|. Whenever we take an absolute value of a function, we take any negative points and make them positive. Essentially, |f(x)| reflects all of the negative points on f(x) across the x-axis. However, points located on the x-axis are unchanged by taking the absolute value, because the absolute value of zero is still going to be zero. In other words, if we take the absolute value of the zeros on f(x), they will still be at the same location. Thus, |f(x)| and f(x) have the same roots.

This leaves the function (f(x))–1. We can set it equal to zero to find its roots. Remember that, in general, _a–_1 = 1/a.

(f(x))–1 = 1/f(x) = 0

Multiply by f(x) on both sides.

1 = 0(f(x)) = 0

Clearly, we have something strange here, because one doesn't equal zero. This tells us that (f(x))–1 can't have the same roots as f(x).

Let's look at an example of why the roots of f(x) and (f(x))–1 won't necessarily be the same. Let f(x) = x. The only root of f(x) would be at the point (0,0). Next, let's look at (f(x))–1 = 1/x. The graph of 1/x won't have any roots, because it has a vertical asymptote at zero (1/x is not defined when x = 0). Thus, f(x) doesn't have the same roots as (f(x))–1. Intuitively, this makes since, because (f(x))–1 essentially takes the reciprocal of the y-values on every point of f(x). But zero doesn't have a reciprocal (because 1/0 isn't defined), so the roots will likely change.

The answer is (f(x))–1 .

How to find f(x)

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