Linear Functions 3 — Heart of Algebra

Questions in this quiz (10)

1. A function $$f$$ is defined for all real numbers such that $$f(a+b)=f(a)+f(b)$$ for all real numbers $$a$$ and $$b$$. If $$f(1)=4$$ and $$f(3x-2)=12x-8,$$ what is the value of $$f(5x+1)$$ in terms of $$x$$?

   • $$20x+4$$
   • $$20x+8$$
   • $$16x+4$$
   • $$24x+4$$

2. A linear function $$f$$ is defined by $$f(x)=mx+b.$$ The graph of $$f$$ passes through the points $$(4,18)$$ and $$(k,6)$$. If the $$y$$-intercept of $$f$$ is $$-6$$, what is the value of $$k$$?

   • $$1$$
   • $$2$$
   • $$3$$
   • $$4$$

3. A linear function $$f$$ is defined by $$f(x)=mx+b,$$ where $$m$$ and $$b$$ are constants. If $$f(4x-5)=20x-9$$ for all values of $$x$$, what is the value of $$2m+b?$$

   • $$26$$
   • $$7$$
   • $$11$$
   • $$13$$

4. A linear function $$f$$ has a nonzero $$y$$-intercept that is three times its $$x$$-intercept. If the graph of $$f$$ passes through the point $$(4,-6)$$, what is the slope of $$f$$?

   • $$-\frac{3}{2}$$
   • $$-2$$
   • $$-\frac{5}{2}$$
   • $$-3$$

5. Two linear functions, $$f$$ and $$g$$, are defined such that $$f(x)=ax+b$$ and $$g(x)=cx+d.$$ It is known that $$f(4)=18$$ and $$g(4)=18.$$ Furthermore, the $$y$$-intercept of $$f$$ is three times the $$y$$-intercept of $$g$$, and the slope of $$f$$ is twice the slope of $$g$$. What is the value of $$x$$ for which $$f(x)=g(x)+8?$$

   • $$\frac{44}{9}$$
   • $$\frac{4}{3}$$
   • $$2$$
   • $$\frac{8}{3}$$

6. A linear function $$f$$ has the property that for every increase of $$3$$ in the value of $$x$$, the value of $$f(x)$$ decreases by $$8$$. If $$f(6)=10,$$ what is the $$x$$-intercept of the graph of $$y=f(x)?$$

   • $$\frac{45}{5}$$
   • $$\frac{39}{5}$$
   • $$\frac{45}{4}$$
   • $$\frac{39}{4}$$

7. A linear function $$f(x)$$ has a slope of $$(3p-2)$$ and a $$y$$-intercept of $$(2p+1)$$, where $$p$$ is a constant. If the graph of $$f(x)$$ passes through the point $$(p,22)$$, what is the positive value of $$p$$?

   • $$\sqrt7$$
   • $$2$$
   • $$3$$
   • $$4$$

8. The linear function $$f$$ is defined such that for every real number $$x$$, $$f(x+5)-f(x)=20.$$ The graph of $$f$$ passes through the point $$(k,4k-3)$$, where $$k$$ is a constant. What is the $$y$$-intercept of the graph of $$f$$?

   • $$-3$$
   • $$1$$
   • $$5$$
   • $$9$$

9. A linear function $$f$$ is defined such that for every real number $$x$$, $$f(x+3)=f(x)+9.$$ If $$f(8)=29,$$ what is the value of $$f(-1)?$$

   • $$2$$
   • $$5$$
   • $$8$$
   • $$11$$

10. A scientist is tracking the growth of two bacteria cultures, Culture A and Culture B. The population of Culture A, in thousands, is modeled by a linear function. On day $$12$$, Culture A has a population of $$42$$, and on day $$36$$, it has a population of $$36$$. The population of Culture B, also in thousands, is modeled by a different linear function. On day $$8$$, Culture B has a population of $$24$$, and on day $$28$$, it has a population of $$84$$. On what day will both cultures have the same population?

    • $$\frac{180}{13}$$
    • $$18$$
    • $$20$$
    • $$24$$