SAT Nonlinear Functions Question 174 | Free SAT Prep

00:00

Try Full-Screen!

Question 174·Hard·Nonlinear Functions

0 / 234

The function $f$ is defined by

f(x)=t\bigl(3^{x}-3^{s}\bigr)+r,

where $t$ and $s$ are nonzero integers and $r$ is a real constant. The functions $g$ and $h$ are equivalent to $f$ .

\text{I. } g(x)=t\cdot 3^x+\bigl(r-t\cdot 3^s\bigr) \\[0.7em] \text{II. } h(x)=t\bigl(3^x-1\bigr)+\bigl(r+t-t\cdot 3^s\bigr)

As $x$ decreases without bound, the graph of $y=f(x)$ approaches a particular $y$ -coordinate. For each equation, consider whether it displays that $y$ -coordinate as a single constant term that can be read directly without combining terms.

Which of the following correctly identifies the equation or equations that do so?

First separate the exponential term from the terms that do not depend on $x$ . Since $3^x$ approaches $0$ as $x$ decreases without bound, the remaining constant gives the y-coordinate approached by the graph. Then check whether each equivalent form shows that exact value as one constant term or requires additional combination of terms.

Hints

Click me!

Use exponential end behavior

Because the base $3$ is greater than $1$ , consider what happens to $3^x$ as $x$ becomes very negative.

Rewrite the original function

Distribute $t$ in $f(x)=t(3^x-3^s)+r$ so that the part containing $3^x$ is separated from the part that does not depend on $x$ .

Inspect the constant terms

For each equivalent form, identify whether the value approached can be read as one $x$ -independent term, rather than found by combining multiple terms.

Desmos Guide

Choose permitted parameter values

Algebra is the most direct method. For a Desmos check, enter t=2, s=1, and r=0.

Graph the equivalent functions

Enter f(x)=t(3^x-3^s)+r, g(x)=t*3^x+(r-t*3^s), and h(x)=t(3^x-1)+(r+t-t*3^s). The three graphs should coincide.

Verify the end behavior

Enter y=r-t*3^s. For the chosen values, this is y=-6. Move left on the graph or inspect a table with large negative $x$ -values to see that all three functions approach this horizontal line. Then compare this value with the constant terms displayed in g(x) and h(x).

Step-by-step Explanation

Determine the y-coordinate approached

Rewrite $f(x)$ by distributing $t$ :

\begin{aligned} f(x)&=t\cdot3^x-t\cdot3^s+r\\ &=t\cdot3^x+\bigl(r-t\cdot3^s\bigr). \end{aligned}

As $x$ decreases without bound, $3^x$ approaches $0$ . Therefore, $t\cdot3^x$ approaches $0$ , and the graph approaches the y-coordinate $r-t\cdot3^s$ .

Check equation I

Equation I is

g(x)=t\cdot3^x+\bigl(r-t\cdot3^s\bigr).

The exponential term $t\cdot3^x$ approaches $0$ as $x$ decreases without bound. The remaining constant term, $r-t\cdot3^s$ , is exactly the y-coordinate approached by the graph. Thus, equation I displays the value directly.

Check equation II

Equation II is

h(x)=t\bigl(3^x-1\bigr)+\bigl(r+t-t\cdot3^s\bigr).

As $x$ decreases without bound, $3^x-1$ approaches $-1$ , so the first term approaches $-t$ . The y-coordinate approached is therefore found only after combining terms:

-t+\bigl(r+t-t\cdot3^s\bigr)=r-t\cdot3^s.

The constant term in equation II is $r+t-t\cdot3^s$ , not $r-t\cdot3^s$ , so equation II does not display the value directly.

Select the applicable equation

Only equation I has a single constant term equal to the y-coordinate approached by the graph as $x$ decreases without bound.

Therefore, the correct answer is I only.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation