The function is defined by Which of the following points lies on the graph of in the -plane?

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SAT Nonlinear Functions Question 65 | Free SAT Prep | Aniko

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Question 65·Medium·Nonlinear Functions

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The function $f$ is defined by

f(x)=\log_{3}(x+6)-2.

Which of the following points $(x,y)$ lies on the graph of $y=f(x)$ in the $xy$ -plane?

For questions asking which point lies on the graph of a given function, use the definition: a point $(x,y)$ is on the graph if $y=f(x)$ . Either plug each option’s $x$ -value into the function and compare the result to the listed $y$ , or, if many choices share the same $y$ -value, set $f(x)$ equal to that $y$ and solve for $x$ directly. Always check domain restrictions (especially for logs and square roots) to quickly eliminate impossible $x$ -values without doing extra arithmetic.

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Connect the function to the points

For a point $(x,y)$ to lie on the graph of $y=f(x)$ , what must be true about $y$ and $f(x)$ ?

Focus on the repeated y-value

Three of the choices have $y=-2$ . Try setting $f(x)=-2$ and solving for $x$ instead of plugging into every choice one by one.

Solve the equation involving the logarithm

If $\log_3(x+6)-2=-2$ , add $2$ to both sides. Then think: what value must $x+6$ be so that its base-3 logarithm is $0$ ?

Remember the domain of a logarithm

The expression inside $\log_3(\,)$ must be positive. Which $x$ -values in the answer choices make $x+6$ positive?

Desmos Guide

Enter the function in Desmos

Type the function exactly as it is: y = log(x+6)/log(3) - 2. (This uses the change-of-base formula to graph $\log_3(x+6)-2$ .)

Check each answer choice numerically

In a separate expression line, type f(x) = log(x+6)/log(3) - 2. Then, for each $x$ -value from the answer choices, type f(-6), f(-5), f(-3), and f(0) and see which output matches the corresponding $y$ -value in the options.

Or check visually with points

You can also add each point manually, like (-6,-2), (-5,-2), (-3,-2), and (0,-1), and see which point lies exactly on the graphed curve of $y=f(x)$ . The point that is on the curve corresponds to the correct answer.

Step-by-step Explanation

Understand what it means for a point to lie on the graph

A point $(x,y)$ lies on the graph of $y=f(x)$ if and only if its coordinates satisfy the equation $y=f(x)$ .

In this problem, that means:

Take the $x$ -value from a choice.
Plug it into $f(x)=\log_3(x+6)-2$ .
See whether the result equals the $y$ -value from that choice.

Use the given y-values to set up an equation

Notice that three answer choices have $y=-2$ and one has $y=-1$ .

Start with the more common $y$ -value, $-2$ . If a point with $y=-2$ is on the graph, its $x$ must satisfy

\log_3(x+6)-2=-2.

Solve this equation for $x$ to find which $x$ -value can go with $y=-2$ on the graph.

Solve $\log_3(x+6)-2=-2$

Add $2$ to both sides:

\log_3(x+6)-2+2=-2+2 \\[0.7em] \log_3(x+6)=0.

Recall: $\log_3(A)=0$ means $3^0=A$ , so

3^0=x+6.

Because $3^0=1$ , we get

x+6=1 \quad \Rightarrow \quad x=1-6=-5.

So the point on the graph with $y=-2$ must have $x=-5$ .

Check domain and verify the correct choice

First, check domain: for $\log_3(x+6)$ , we need $x+6>0$ , so $x>-6$ . That means $x=-6$ is not even in the domain, so choice $(-6,-2)$ cannot lie on the graph.

From Step 3, the only $x$ that makes $f(x)=-2$ is $x=-5$ , giving the point $(-5,-2)$ .

Thus, the point that lies on the graph of $y=f(x)$ is $(-5,-2)$ , which is answer choice B.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation